## Abstract

A circular zero-time-averaged power component, coupling the forward (dielectric) and backward (metal) power channels of Surface Plasmon Polaritons (SPPs), is shown to be the core ingredient for the slow-light characteristic of SPPs at the surface plasmon frequency, for both a lossless and lossy metal. Additional slow-light regimes emerging in configurations where few SPPs are strongly coupled, such as in a narrow plasmonic gap and slab, forming local extrema of the dispersion curve (branch points for positive and negative index branches), are also propelled by the circular motion of the plasmonic power.

©2010 Optical Society of America

## 1. Introduction

Metallic nano-particles and nano-structures supporting plasmon-polaritons, are key ingredient in the fields of nanoplasmonics and metamaterials at the visible and near IR ranges. The very large collective dipole oscillations of the metallic plasma electrons are one origin of the anomalous dispersion in plasmonic waveguide structures, while the waveguide geometry is another source which is of no less importance as will be further discussed. As a result, surface plasmon polaritons exhibit a variety of exceptional characteristics such as slow wave [1,2], possibly several regimes of slow light and 'stopped light' [3–5], backward waves [6–10], fast light [11], etc. These phenomena are the facilitators of nano (sub-wavelength) guiding [12], efficient power concentration into the nanoscale regime [13], cavities with extremely small modal volumes [14,15], etc. The exact resolution of the physical mechanisms for this extraordinary variety is the purpose of this paper. Naively, one may attribute all of these phenomena to an advanced storage of the electric field energy in the dark polariton oscillations (the collective plasma electrons in our case) similar to the explanation of light slowing and stopping in atomic medium [16], however the explanation put forth herein paints a rather different picture in plasmonic waveguiding structures.

A single metal-dielectric interface supports a TM polarized surface mode – the Surface Plasmon Polariton (SPP) [1]. The slow wave behavior of this mode (slow phase velocity) becomes prominent as its frequency approaches the characteristic surface plasmon frequency and it is accompanied by slow-light characteristics (reduced dispersion slope or group velocity). It should be emphasized that all SPP supporting structures are viable only at frequencies deep within the electromagnetic band gap of the bulk metal, where the predominant electromagnetic field is longitudinal and only the dipoles on the interface allow 'transverse' electromagnetic field to propagate. More elaborated structures, such as plasmonic gap or slab waveguides which are comprised of two metal-dielectric interfaces, may also support negative effective index mode (backward-waves) [6], and few regimes of slow-stopped light. Near the plasmon polariton band-gap frequencies (the gap between the existence of propagating bulk metal plasmons and propagating SPPs) the metal loss enables also 'fast light' (negative dispersion slope) characteristics [11,17] – the latter is not discussed here in detail.

Many of these unique dispersion phenomena can be readily attributed to the coupled counter-propagation in the metal and dielectric parts exhibiting counter power flow of the SPP [18], stemming from the negative sign of permittivity in the metal that reverses its longitudinal power component – counter propagating to that in the dielectric. An essential coupling between these seemingly two distinct power flow channels is a strange notion for an eigen-mode propagation within a *z*-invariant structure, in contrast to a structure supporting space harmonics where forward and backwards power propagation is coupled by a periodic index perturbation – it is nevertheless fundamental to our interpretation. The basic coupling is performed, as will be discussed, by a zero time average transverse power flow from the dielectric to the metal and vice versa that completes a power flow cycle. The other two parts of the cycle (longitudinal power flow in the dielectric and metal) include also real (non-zero time average) power flows.

The wave dispersion can be connected to the energy flow when losses are not extremely large (i.e. figure of merit |Re{*β*}/Im{*β*}|>1 where *β* is the propagation constant). In such cases the energy velocity *v _{e}* is related to the power flow

*S*and the stored energy

*U*by

*v*=

_{e}*S*/

*U*. One can reduce the energy velocity (slow light) by increasing the stored energy in the medium, e.g. as is done in the experiments of Hau [16], or by reducing the power flow which is performed by partially back coupling the propagating power – the embedded mechanism of light slowing in cavity arrays [19], and in photonic band gap structures [20]. In plasmonics the built-in circular power coupling is the major mechanisms for light slowing, stopping and even for backward waves (negative index) modes.

## 2. Circular power flow of SPPs and slow-light

The fact that the time-averaged power flow on both sides of a metal-dielectric interface propagates at opposite directions, forming the backward and forward channels respectively, implies some mechanism of coupling between the two channels. The first paper presenting the modes of a plasma slab [21], conjectured that such coupling necessitates metal loss which tilts the phase front and delivers real power from the lossless dielectric to the metal. As discussed here, this is not the core mechanism and coupling exists also for a lossless case.

While in a structure supporting space harmonics forward to backwards power coupling is generated by a periodic index perturbation, in the *z*-invariant metal-dielectric interface coupling is generated by the harmonic surface charge distribution. The latter is inducing a vertical (across the interface) instantaneous, zero-time-averaged power flow component that periodically transfers power from one channel to the other and then vice versa – completing a circular power flow between the two channels. The circular power trajectories propagate at the phase velocity, following the propagation of the surface charge distribution, while the energy velocity is determined by the net power flow (the difference in power flow between the two channels). Thus, slow-light in SPPs is achieved via this cyclic power flow coupling mechanism.

A quantitative description of the power coupling mechanism is derived. A SPP propagating along the *z* direction of a metal (*x*>0) dielectric (*x*<0) interface, with metal permittivity given by the Drude form

*ω*is the angular plasma frequency (in this paper we use

_{p}*ω*= 1.37x10

_{p}^{16}for gold),

*γ*the electron scattering rate (we use

*γ*= 4.05x10

^{13}for gold),

*ε*

_{0}the permittivity of vacuum and

*ω*the angular frequency. The dielectric permittivity

*ε*is assumed frequency independent.

_{D}The TM SPP wave has an *H*-field (phasor) of the form

*β*

^{2}= (

*ω*/

*c*)

^{2}

*∙ε*∕

_{D}ε_{M}*(ε*), the decay coefficient in the dielectric and metal are given by

_{D}+ ε_{M}*κ*

^{2}

*=*

_{D,M}*β*

^{2}– (

*ω*/

*c*)

^{2}

*ε*

_{D}_{,M}respectively, and

*c*is the vacuum light velocity.

The time-dependent Poynting vector, derived from Eq. (2), is (for brevity we assume here a lossless case; we address losses in section 6):

*x*) and longitudinal (

*z*)

*E*-field amplitudes at the interface, and square brackets denote the

*x*and

*z*components of a vector. The first term on the RHS of Eq. (3a) is the time-averaged power flow, given by Eq. (3b), which has opposite signs in the dielectric and metal. The second term is a zero-time-averaged power flow component which represents coupling between the forward and backward channels (Fig. 1 ). The instantaneous power trajectory resulting from this coupling term propagates at the phase velocity along the interface.

The relation between this power trajectory and the slow-light characteristic of SPPs can be traced to the circular motion of power delaying energy propagation; the more prominent the circular motion is, the slower the energy will propagate along the interface.

To quantify this assertion we define two coupling coefficients representing the exchange of power between the two channels: forward (D) to backward (M) *C _{D}*

_{→}

*and*

_{M}*C*

_{M}_{→}

*, as the ratio of power transferred to the other channel to that flowing at a given channel. The instantaneous power flowing in each of the channels is the integrated power in each half plane while the coupled power between channels is the integrated power crossing the interface in half a spatial cycle:*

_{D}We also use the rotor of the power flow at the interface, as a measure for circularity of the power flow trajectory (harmonic time dependence term is omitted):

Next we examine the relationship of these measures to the net power flow of the SPP (total power crossing a given plane vertical to the interface), given by:

Figure 2a
shows the frequency dependence of the circularity (Eq. (5)) and net power flow (Eq. (6)), and Fig. 2b shows the coupling coefficients (Eq. (4)) *C _{D}*

_{→}

*and*

_{M}*C*

_{M}_{→}

*which are also equal to the square of the*

_{D}*E*-field amplitude ratio at both ends of the interface. As the frequency approaches the surface plasmon frequency, where slow-light is most prominent (net power flow vanishes), the circularity measure is dramatically enhanced and both coupling coefficients converge to 1 indicating that all power flowing forward is completely coupled backwards, closing a circular loop with no net energy propagation – hence stopped-light.

These results corroborate our assertion that the forward-to-backward coupling, caused by the harmonic surface charge distribution, is the underlying mechanism behind the slow-light property of SPP, exhibited mainly at frequencies approaching the characteristic surface plasmon frequency.

## 3. Plasmonic slab

In a planar slab configuration – a thin metal film of thickness *d _{D}* embedded in a dielectric (Fig. 3a
), the evanescent coupling of the two adjacent SPPs, results in a more complex circular power flow, giving rise to additional ‘stopped-light’ points. The well known dispersion curves of the confined plasmonic modes – the anti-symmetric (TM

_{0}

^{S}) and the symmetric (TM

_{1}

^{S})

*H*-fields are depicted in Fig. 3b (for various film thicknesses). As seen in the figure, when film thickness is substantially decreased, the strong coupling of SPPs results in a significant mode splitting such that the dispersion curve of the symmetric slab mode penetrates into the band gap. Since its asymptotic behavior at the surface plasmon frequency is the same as that of a single SPP, it must exhibit at least two local extrema as it asymptotically converges to the SPP curve (Figs. 3b, 3c). These extrema are in fact branch points (as is shown in section 6), such that the dispersion curve segmented between these points, exhibiting a negative slope, is actually located at the reciprocal negative propagation constant side (Fig. 3c) – having negative effective index but a positive group velocity [16]. We continue to use the combined dispersion in the positive

*β*values throughout this paper for visual convenience.

To study the cyclic power coupling near these extrema we calculate the time dependent Poynting vector for TM_{1}
^{S} (for TM_{0}
^{S}, ${f}_{T{M}_{0}^{S}}\left(x\right)$ and ${f}_{T{M}_{1}^{S}}\left(x\right)$ should be interchanged):

*E*-field amplitudes are ${E}_{\perp}^{S}\left(x\right)=\beta /\omega \epsilon \left(x\right)$ and ${E}_{\parallel}^{S}=-{\kappa}_{D}/\omega {\epsilon}_{D}$ , ${f}_{T{M}_{0}^{S}}\left(x\right)$ and ${f}_{T{M}_{1}^{S}}\left(x\right)$ are the

*H*-field profiles of TM

_{0}

^{S}and TM

_{1}

^{S}respectively, and the time-averaged power flow term on the RHS of Eq. (7a) is given by Eq. (7b).

The backward channel in the metal film and the two forward channels in the dielectric surrounding are coupled by the circular zero-time-average term, generating two engaged closed-loops power flows crossing the two interfaces (Fig. 5 ) with the trajectory propagating at the phase velocity. The net forward power flow of each of the slab modes is given by:

_{0}

^{S}and TM

_{1}

^{S}respectively.

The power and coupling contain now a hyperbolic term which stems from the SPP coupling (via the metal film) which at *κ _{M}d_{M}*→

*∞*decouple to two SPPs – thus retrieving the asymptotic ‘stopped-light’ characteristics of the previous section at the SPP frequency. TM

_{0}

^{S}does not exhibit additional nulls in the net power flow but for the TM

_{1}

^{S}, net power propagation halts when

The LHS term in Eq. (10) is large at low frequencies and vanishes as (*ω* – *ω _{SPP}*)

^{2}at frequencies approaching the surface plasmon frequency. The RHS term is (

*k*)/sinh(

_{p}d_{M}*k*) at low frequencies (

_{p}d_{M}*k*=

_{p}*ω*/

_{p}*c*) and exponentially vanishes at the surface plasmon frequency. It follows that below a threshold thickness, as the RHS becomes larger, exactly two solutions (frequencies for which Eq. (10) holds) exist. E.g., for film thicknesses smaller than 32nm at the parameters of Fig. 4b , local maximum and minimum in the dispersion curve exist.

The coupling coefficients *C _{D}*

_{→}

*and*

_{M}*C*

_{M}_{→D}at the local extrema coincide (Fig. 4c), indicating the same amount of power flowing through the forward and backward channels – hence stopped-light as net power vanishes (Fig. 4a). However, unlike the behavior exhibited near the characteristic surface plasmon frequency, the coefficients are not equal to 1.

The power trajectory for the negative index branch is discussed in section 5.

## 4. Plasmonic gap

In a plasmonic gap structure, comprised of a dielectric layer of width *d _{D}* between two plasmonic metals (Fig. 6a
), the lower order modes are the symmetric (TM

_{0}

^{G}) and anti-symmetric (TM

_{1}

^{G})

*H*-fields. The TM

_{1}

^{G}mode exhibits a cut-off (Fig. 6b), where the effective mode index is zero and it is also a local extremum of the dispersion curve – namely a point of ‘stopped’ light. Reducing the gap width, the TM

_{1}

^{G}cut-off frequency is enhanced (enhanced mode splitting) and eventually becomes a local maximum rather than a local minimum (Fig. 7b the transition from

*d*= 60nm to 55nm). Since the dispersion curve converges to the SPP curve, essential local minimum is generated as well, with ‘stopped light’ characteristics.

_{D}The net power flow and coupling coefficients are obtained by interchanging *ε _{D}*↔

*ε*,

_{M}*κ*↔

_{D}*κ*,

_{M}*d*↔

_{D}*d*, TM

_{M}_{0}↔TM

_{1}in the corresponding expressions derived for the slab configuration in Eqs. (8) and (9), yielding:

_{0}

^{G}and TM

_{1}

^{G}respectively. The expression for the Poynting vector has the same structure as Eq. (7), hence the circular motion of power, comprised of two engaged closed-loops crossing the two interfaces, is in effect for both gap modes as well.

As in the previous section, at *κ _{D}d_{D}*→

*∞*the asymptotic stopped-light at the SPP frequency is retrieved. In addition, there is always one extremum for TM

_{1}

^{G}at

*β*= 0, while the other exists only below a certain threshold for

*d*given by:

_{D}The coupling coefficients *C _{D}*

_{→}

*and*

_{M}*C*

_{M}_{→}

*at the local minimum coincide (Fig. 7c), indicating the same amount of power is flowing through the forward and backward channels – hence stopped-light as net power vanishes (Fig. 7a).*

_{D}While the stopped light in all the previously discussed extremum points was due to equalizing the power in the two channels, the stopped light exhibited in the cut-off frequency (*β* = 0), is due to halting of power propagation in both channels (Fig. 7a). Here the power is completely stored reactively in the dipolar charges on the interfaces. This can be deduced by an examination of the coupling coefficients, both of which rapidly increase as frequency approaches the cut-off (Fig. 7c) – indicating that the power in both the forward and backward channels vanishes, while power only flows upward and downward periodically. The mode does not propagate and its energy is transferred to the stationary surface dipoles.

## 5. Cyclic power flow at backward-waves branch

Backward waves (negative phase velocities) in both gap and slab configurations are the characteristic modes of the branch between two local extrema of the dispersion curve where more power propagates in the metal than in the dielectric. The typical power flow still exhibits the circular motion but since the trajectory propagates as a whole at the phase velocity, in the negative index branch this trajectory is counter propagating to the net power flow (the latter is always positive), while co-directional for the positive index branch. This can be seen in the time snapshot of Fig. 8a and 8c (Media 1) for the positive index branch, and in Fig. 8b and 8d (Media 2) for the negative index branch of a plasmonic gap.

## 6. The effect of metal loss on the circular power flow

It may be questioned if the above analysis is valid also for the actual lossy metals. As will be shown here, the previously discussed results are almost unchanged. Using as an example a lossy metal with dispersion according to the Drude form Eq. (1), the effects of losses on the circular motion of power and the related stopped-light in a gap configuration are studied. The time dependent Poynting vector for TM_{1}
^{G} can be written as:

*E*-field amplitudes are ${E}_{\perp}^{G}\left(x\right)=\beta /\omega \epsilon \left(x\right)$ and ${E}_{\parallel}^{G}=-{\kappa}_{M}/\omega {\epsilon}_{M}$ , ${f}_{T{M}_{0}^{G}}$ and ${f}_{T{M}_{1}^{G}}$ are the

*H*-field profiles of TM

_{0}

^{G}and TM

_{1}

^{G}respectively (

*x*-dependence omitted for brevity), the time-averaged power flow on the RHS of Eq. (14a) is given by Eq. (14b), and asterisk denotes complex conjugation (for a slab one should interchange

*ε*↔

_{D}*ε*,

_{M}*κ*↔

_{D}*κ*,

_{M}*d*↔

_{D}*d*, TM

_{M}_{0}↔TM

_{1}; for a SPP both ${f}_{T{M}_{0}^{G}}$ and ${f}_{T{M}_{1}^{G}}$ should be replaced by the SPP

*H*-field profile).

As seen in Eq. (14), even though there is an additional *x*-component of the average power flow from the dielectric into the metal to compensate for the metal losses, the circular motion of the zero- average power is retained as the dominant mechanism in coupling the forward and backward channels whenever the figure of merit |*β _{r} /β_{i}*|>1, i.e. as long as the mode is actually propagating.

A delicate point is the actual wave characteristics near the 'stopped light' local extrema – the branch points between the positive and negative index branches. It is seen that in the lossy case these points are only approximately reached as the two branches split apart (dashed and solid curves Fig. 9b
). However, at all frequencies where the dispersion branches are related to actually propagating modes (|*β _{r} /β_{i}*|>1 in Fig. 9a and 9c), the dispersion curve very closely coincides with that of the lossless case (dashed black curve in Fig. 9b). Hence, although slow-light is achieved when approaching the branch point, actual stopping of light is not feasible.

Similar reasoning applies for the positive index branch at the SPP frequency, where the dispersion curve of the positive index branch (and also the SPP curve in dotted purple) is folded back towards the band gap and deviates from the lossless curve, as losses become prominent and the branch is no longer considered propagating.

## 7. Summary

In this paper we have shown that the slow-light characteristic of SPPs, prominent at the surface plasmon frequency, is brought about by a zero-time-averaged circular motion of power coupling the seemingly separate forward (dielectric) and backward (metal) channels. Even when losses are considered, and a transverse component of the average power flow exists, the circular motion remains the dominant coupling mechanism as long as the mode is actually a propagating mode.

Additional stopped-light points may be achieved at local extrema of the dispersion curve in planar configurations where SPP evanescent coupling is sufficiently strong (such as a plasmonic slab or gap, below a certain threshold thickness). These extrema are in fact branch points separating negative and positive index branches. Both branches are characterized by the same circular motion of power but with more power flowing in either the metal (negative index branch) or the dielectric (positive index branch) channels. Hence, the instantaneous power trajectory resulting from the circular motion of power, which propagates at the phase velocity along the interfaces, is either counter-propagating or co-propagating to the mode energy in the negative and positive index branches respectively. When losses are considered, slow-light is still exhibited in the frequencies around these branch points even though stopped-light in the actual branch points themselves is prohibited. The same applies at the surface plasmon frequency.

In a gap configuration an additional stopped-light mechanism, of the transfer of the mode energy to surface dipoles, is available at the cut-off frequency for TM_{1}
^{G}.

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