Circular motion of electromagnetic power shaping the dispersion of Surface Plasmon Polaritons

doi:10.1364/OE.18.025861

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Circular motion of electromagnetic power shaping the dispersion of Surface Plasmon Polaritons

Gilad Rosenblatt, Eyal Feigenbaum, and Meir Orenstein »View Author Affiliations

Optics Express, Vol. 18, Issue 25, pp. 25861-25872 (2010)
http://dx.doi.org/10.1364/OE.18.025861

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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.

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

R. H. Ritchie, “Plasma Losses by Fast Electrons in Thin Films,” Phys. Rev. 106(5), 874–881 (1957). [CrossRef]

M. Sandtke and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photonics 1(10), 573–576 (2007). [CrossRef]

], possibly several regimes of slow light and 'stopped light' [3

M. Orenstein, “Slow Light by Slow Waves: Plasmonics for Light Halting,” Topical Meeting on Slow and Fast Light 2007, Salt-Lake City Utah, USA (2007).

–5

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007). [CrossRef] [PubMed]

], backward waves [6

A. A. Oliner and T. Tamir, “Backward waves on Isotropic Plasma Slabs,” J. Appl. Phys. 33(1), 231–233 (1962). [CrossRef]

–10

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9(5), 407–412 (2010). [CrossRef] [PubMed]

], fast light [11

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]

], etc. These phenomena are the facilitators of nano (sub-wavelength) guiding [12

E. Feigenbaum and M. Orenstein, “Modeling of Complementary (Void) Plasmon Waveguiding,” JLT 25, 2547–2562 (2007).

], efficient power concentration into the nanoscale regime [13

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 (2004). [CrossRef] [PubMed]

], cavities with extremely small modal volumes [14

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]

,15

E. Feigenbaum and M. Orenstein, “Ultrasmall volume plasmons, yet with complete retardation effects,” Phys. Rev. Lett. 101(16), 163902 (2008). [CrossRef] [PubMed]

], 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

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001). [CrossRef] [PubMed]

], 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

R. H. Ritchie, “Plasma Losses by Fast Electrons in Thin Films,” Phys. Rev. 106(5), 874–881 (1957). [CrossRef]

]. 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

A. A. Oliner and T. Tamir, “Backward waves on Isotropic Plasma Slabs,” J. Appl. Phys. 33(1), 231–233 (1962). [CrossRef]

], 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

E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative dispersion: a backward wave or fast light? Nanoplasmonic examples,” Opt. Express 17(21), 18934–18939 (2009). [CrossRef]

] – 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

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67(3), 035109 (2003). [CrossRef]

], 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

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef]

], and in photonic band gap structures [20

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87(25), 253902 (2001). [CrossRef] [PubMed]

]. 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

T. Tamir and A. A. Oliner, “The Spectrum of Electromagnetic Waves Guided by a Plasma Layer,” Proc. IEEE 51(2), 317–332 (1963). [CrossRef]

], 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

ε_{M} (ω) = ε_{0} (1 - \frac{ω_{p}^{2}}{ω (ω + j γ)})

(1)

where ω_p is the angular plasma frequency (in this paper we use ω_p = 1.37x10¹⁶ for gold), γ the electron scattering rate (we use γ = 4.05x10¹³ for gold), ε ₀ the permittivity of vacuum and ω the angular frequency. The dielectric permittivity ε_D is assumed frequency independent.

The TM SPP wave has an H-field (phasor) of the form

H_{y} (x, z) = e^{- j β z} \cdot f_{S P P} (x) ≜ e^{- j β z} \cdot {\begin{matrix} e^{κ_{M} x}, x < 0 \\ e^{- κ_{D} x}, x > 0 \end{matrix}

(2)

where the propagation constant is β ² = (ω/c)² ∙ε_Dε_M ∕(ε_D + ε_M ), the decay coefficient in the dielectric and metal are given by 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):

[S_{x}, S_{z}] = [{\bar{S}}_{x}, {\bar{S}}_{z}] + [E_{∥}^{S P P} \cdot \sin (2 ϕ), E_{⊥}^{S P P} (x) \cdot \cos (2 ϕ)] \cdot f_{S P P}^{2} (x) / 2

(3a)

[{\bar{S}}_{x}, {\bar{S}}_{z}] = [0, E_{⊥}^{S P P} (x) / 2] \cdot f_{S P P}^{2} (x)

(3b)

where

ϕ = ω t - β z

is the phase,

E_{⊥}^{S P P} (x) = β / ω ε (x)

and

E_{∥}^{S P P} = κ_{D} / ω ε_{D}

are the transverse (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.

Fig. 1 Circular power flow of a SPP propagating along a lossless gold (x<0) air (x>0) interface for vacuum wavelength λ ₀ = 216nm; Poynting vector (small arrows) and its magnitude (contour).The circular motion is illustrated by the thick arrows inset.

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 _→ _M and C_M _→ _D , 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:

C_{D \to M} ≜ \frac{P_{↓}}{P_{\to}} = {(\frac{κ_{D}}{β})}^{2}

(4a)

C_{M \to D} ≜ \frac{P_{↑}}{P_{\leftarrow}} = {(\frac{κ_{M}}{β})}^{2}

(4b)

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):

{| \nabla \times \vec{S} |}_{x \to 0} = \frac{β κ_{D}}{ω ε_{D}}

(5)

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:

P_{S P P} ≜ \int_{- \infty}^{\infty} [S_{x}, S_{z}] \cdot \hat{z} d x = \frac{β}{ω κ_{M} ε_{M}} (1 - \frac{κ_{M}^{2}}{κ_{D}^{2}}) \cos^{2} (ω t - β z)

(6)

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 _→ _M and C_M _→ _D which are also equal to the square of the 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.

Fig. 2 Frequency dependence of (a) the circularity measure (green) and net power flow (purple), and (b) the coupling coefficients C_D _→ _M (blue) and C_M _→ _D (red) in logarithmic scale, of a SPP propagating along a lossless gold-air interface. SPP frequency – dashed gray lines.

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 (TM₀ ^S) and the symmetric (TM₁ ^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.

Fig. 3 (a) Schematics of a metal slab and the H-field distribution of the modes, (b) dispersion curves of TM₀ ^S (solid) and TM₁ ^S (dashed) for lossless gold slabs in air at various thicknesses, and (c) a close-up on the positive (green) and negative (red) index branches for the dispersion curve of TM₁ ^S at their actual position (power is propagating to positive z direction). Black arrows in (b) indicate the direction of stronger SPP coupling (decreasing d_M ). The SPP dispersion curve – dashed black; the light line in (b) – black line; SPP frequency – dashed gray lines.

To study the cyclic power coupling near these extrema we calculate the time dependent Poynting vector for TM₁ ^S (for TM₀ ^S,

f_{T M_{0}^{S}} (x)

and

f_{T M_{1}^{S}} (x)

should be interchanged):

{[S_{x}, S_{z}]}_{T M_{1}^{S}} = {[{\bar{S}}_{x}, {\bar{S}}_{z}]}_{T M_{1}^{S}} + [E_{∥}^{S} \sin (2 ϕ) f_{T M_{0}^{S}} (x) f_{T M_{1}^{S}} (x), E_{⊥}^{S} (x) \cos (2 ϕ) f_{T M_{1}^{S}}^{2} (x)] / 2

(7a)

{[{\bar{S}}_{x}, {\bar{S}}_{z}]}_{T M_{1}^{S}} = [0, E_{⊥}^{S} (x)] \cdot f_{T M_{1}^{S}}^{2} (x) / 2

(7b)

where the E-field amplitudes are

E_{⊥}^{S} (x) = β / ω ε (x)

and

E_{∥}^{S} = - κ_{D} / ω ε_{D}

f_{T M_{0}^{S}} (x)

and

f_{T M_{1}^{S}} (x)

are the H-field profiles of TM₀ ^S and TM₁ ^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:

P (ω) = \frac{β}{ω κ_{D} ε_{D}} (1 - \frac{κ_{D}^{2}}{κ_{M}^{2}} (1 \mp \frac{κ_{M} d_{M}}{\sinh (κ_{M} d_{M})})) \cos^{2} (ω t - β z)

(8)

and the coupling coefficients are given by:

C_{D \to M} ≜ \frac{P_{↓}}{P_{\to}} = {(\frac{κ_{D}}{β})}^{2}

(9a)

C_{M \to D} ≜ \frac{P_{↑}}{P_{\leftarrow}} = {(\frac{κ_{M}}{β})}^{2} {(1 \mp \frac{κ_{M} d_{M}}{\sinh (κ_{M} d_{M})})}^{- 1}

(9b)

where minus and plus signs in Eqs. (8) and (9b) correspond to TM₀ ^S and TM₁ ^S respectively.

Fig. 5 Poynting vector (arrows) and its magnitude (contour) for TM₁ ^S of a lossless gold slab (in air) of thickness d_M = 30nm, at frequency corresponding to (a) the local maximum, and (b) near the characteristic SPP frequency.

The power and coupling contain now a hyperbolic term which stems from the SPP coupling (via the metal film) which at κ_Md_M →∞ decouple to two SPPs – thus retrieving the asymptotic ‘stopped-light’ characteristics of the previous section at the SPP frequency. TM₀ ^S does not exhibit additional nulls in the net power flow but for the TM₁ ^S, net power propagation halts when

\frac{κ_{M}^{2}}{κ_{D}^{2}} - 1 = \frac{κ_{M} d_{M}}{\sinh (κ_{M} d_{M})}

(10)

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_pd_M )/sinh(k_pd_M ) at low frequencies (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.

Fig. 4 (a) Net power flow, (b) dispersion curve, and (c) coupling coefficients (in logarithmic scale) C_D _→ _M (dashed) and C_M _→ _D (solid), for TM₁ ^S for lossless gold slabs in air at various thicknesses. Black rectangles in (c) indicate coincidence of the coupling coefficients at the extrema for the red curves. The SPP dispersion curve in (b) – dashed black; SPP frequency – dashed gray lines; vertical black lines in (a) and (c) indicates 0 and 1 respectively.

The coupling coefficients C_D _→ _M and 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 (TM₀ ^G) and anti-symmetric (TM₁ ^G) H-fields. The TM₁ ^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 TM₁ ^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_D = 60nm to 55nm). Since the dispersion curve converges to the SPP curve, essential local minimum is generated as well, with ‘stopped light’ characteristics.

Fig. 6 Schematics of a gap of thickness d_D and the H-field of the gap modes, and (b) the dispersion curves of TM₀ ^G (solid) and TM₁ ^G (dashed) for air gaps (in lossless gold) at various widths. SPP dispersion curve – dashed black; light line – black line; SPP frequency – dashed gray line.

Fig. 7 (a) Net power flow, (b) dispersion curve, and (c) coupling coefficients (logarithmic scale) C_D _→ _M (dashed) and C_M _→ _D (solid), for TM₁ ^G of air gaps (in lossless gold) at various widths. Black rectangles in (c) indicate coincidence of the coupling coefficients at the extremum for the red curves; SPP frequency – dashed gray lines; vertical black lines in (a) and (c) indicates 0 and 1 respectively.

The net power flow and coupling coefficients are obtained by interchanging ε_D ↔ε_M , κ_D ↔κ_M , d_D ↔d_M , TM₀↔TM₁ in the corresponding expressions derived for the slab configuration in Eqs. (8) and (9), yielding:

P (ω) = \frac{β}{ω κ_{M} ε_{M}} (1 - \frac{κ_{M}^{2}}{κ_{D}^{2}} (1 \pm \frac{κ_{D} d_{D}}{\sinh (κ_{D} d_{D})})) \cos^{2} (ω t - β z)

(11)

C_{D \to M} ≜ \frac{P_{↓}}{P_{\to}} = {(\frac{κ_{D}}{β})}^{2} {(1 \pm \frac{κ_{D} d_{D}}{\sinh (κ_{D} d_{D})})}^{- 1}

(12a)

C_{M \to D} ≜ \frac{P_{↑}}{P_{\leftarrow}} = {(\frac{κ_{M}}{β})}^{2}

(12b)

where the plus and minus signs in Eqs. (11) and (12a) correspond to TM₀ ^G and TM₁ ^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_Dd_D →∞ the asymptotic stopped-light at the SPP frequency is retrieved. In addition, there is always one extremum for TM₁ ^G at β = 0, while the other exists only below a certain threshold for d_D given by:

1 - \frac{κ_{D}^{2}}{κ_{M}^{2}} = \frac{κ_{D} d_{D}}{\sinh (κ_{D} d_{D})}

(13)

e.g. for a gap width smaller than 57nm in the parameters of Fig. 7b, a local minimum of the dispersion curve is generated (solution for Eq. (13) exists).

The coupling coefficients C_D _→ _M and C_M _→ _D 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).

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.

Fig. 8 Snapshots of the Poynting vector (small arrows) and magnitude (contours) of TM₁ ^G for an air gap (in lossy gold) of width d_D = 50nm at frequency ω = 0.682ω_p and at times t = 0 (a,b) and t = π/4ω (c,d), for the positive (a,c, Media 1) and negative (b,d, Media 2) index branches. Propagation of the circular power trajectory (at phase velocity) is tracked by green and red ellipses for the positive and negative index branches respectively. The overall power flowing through the dielectric and metal channels is illustrated by the magnitude of the thick green and red arrows respectively, while the direction of the net power flow is positive for both branches (the direction of decreasing magnitude).

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 TM₁ ^G can be written as:

{[S_{x}, S_{z}]}_{T M_{1}^{S}} = {[{\bar{S}}_{x}, {\bar{S}}_{z}]}_{T M_{1}^{S}} + [Re {E_{∥}^{G} f_{T M_{0}^{G}} f_{T M_{1}^{G}}} \sin (2 ϕ), Re {E_{⊥}^{G} f_{T M_{1}^{G}}^{2}} \cos (2 ϕ)] \cdot e^{2 β_{i} z} / 2

- [Im {E_{∥}^{G} f_{T M_{0}^{G}} f_{T M_{1}^{G}}} \sin (2 ϕ), Im {E_{⊥}^{G} f_{T M_{1}^{G}}^{2}} \cos (2 ϕ)] \cdot e^{2 β_{i} z} / 2

(14a)

{[{\bar{S}}_{x}, {\bar{S}}_{z}]}_{T M_{1}^{S}} = [Im {{(E_{∥}^{G} f_{T M_{0}^{G}})}^{*} f_{T M_{1}^{G}}}, Re {{(E_{⊥}^{G} f_{T M_{1}^{G}})}^{*} f_{T M_{1}^{G}}}] \cdot e^{2 β_{i} z} / 2

(14b)

where

ϕ = ω t - β_{r} z

β = β_{r} + j β_{i}

(

β_{i} < 0

), the E-field amplitudes are

E_{⊥}^{G} (x) = β / ω ε (x)

and

E_{∥}^{G} = - κ_{M} / ω ε_{M}

f_{T M_{0}^{G}}

and

f_{T M_{1}^{G}}

are the H-field profiles of TM₀ ^G and TM₁ ^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_M , TM₀↔TM₁; 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.

Fig. 9 The (a,c) imaginary and (b) real parts the propagation constant β of TM₁ ^G for air gaps (in lossy gold) at various widths. The curves of the positive and negative branches are in solid and dashed respectively. The SPP dispersion curve (with losses) is in dotted purple. A zoomed section of the branch point for the red curves in (b) is presented inset for clarity. The black arrows in (a) and (c) indicate the direction of increasing losses.

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 TM₁ ^G.

References and links

1.	R. H. Ritchie, “Plasma Losses by Fast Electrons in Thin Films,” Phys. Rev. 106(5), 874–881 (1957). [CrossRef]
2.	M. Sandtke and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photonics 1(10), 573–576 (2007). [CrossRef]
3.	M. Orenstein, “Slow Light by Slow Waves: Plasmonics for Light Halting,” Topical Meeting on Slow and Fast Light 2007, Salt-Lake City Utah, USA (2007).
4.	A. A. Govyadinov and V. A. Podolskiy, “Gain-assisted slow to superluminal group velocity manipulation in nanowaveguides,” Phys. Rev. Lett. 97(22), 223902 (2006). [CrossRef] [PubMed]
5.	K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007). [CrossRef] [PubMed]
6.	A. A. Oliner and T. Tamir, “Backward waves on Isotropic Plasma Slabs,” J. Appl. Phys. 33(1), 231–233 (1962). [CrossRef]
7.	H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006). [CrossRef] [PubMed]
8.	H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007). [CrossRef] [PubMed]
9.	V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007). [CrossRef]
10.	S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9(5), 407–412 (2010). [CrossRef] [PubMed]
11.	G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]
12.	E. Feigenbaum and M. Orenstein, “Modeling of Complementary (Void) Plasmon Waveguiding,” JLT 25, 2547–2562 (2007).
13.	M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 (2004). [CrossRef] [PubMed]
14.	H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]
15.	E. Feigenbaum and M. Orenstein, “Ultrasmall volume plasmons, yet with complete retardation effects,” Phys. Rev. Lett. 101(16), 163902 (2008). [CrossRef] [PubMed]
16.	C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001). [CrossRef] [PubMed]
17.	E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative dispersion: a backward wave or fast light? Nanoplasmonic examples,” Opt. Express 17(21), 18934–18939 (2009). [CrossRef]
18.	G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67(3), 035109 (2003). [CrossRef]
19.	A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef]
20.	M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87(25), 253902 (2001). [CrossRef] [PubMed]
21.	T. Tamir and A. A. Oliner, “The Spectrum of Electromagnetic Waves Guided by a Plasma Layer,” Proc. IEEE 51(2), 317–332 (1963). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(350.4238) Other areas of optics : Nanophotonics and photonic crystals

ToC Category:
Optics at Surfaces

History
Original Manuscript: August 23, 2010
Revised Manuscript: October 22, 2010
Manuscript Accepted: October 28, 2010
Published: November 25, 2010

Citation
Gilad Rosenblatt, Eyal Feigenbaum, and Meir Orenstein, "Circular motion of electromagnetic power shaping the dispersion of Surface Plasmon Polaritons," Opt. Express 18, 25861-25872 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25861

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References

R. H. Ritchie, “Plasma Losses by Fast Electrons in Thin Films,” Phys. Rev. 106(5), 874–881 (1957). [CrossRef]
M. Sandtke and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photonics 1(10), 573–576 (2007). [CrossRef]
M. Orenstein, “Slow Light by Slow Waves: Plasmonics for Light Halting,” Topical Meeting on Slow and Fast Light 2007, Salt-Lake City Utah, USA (2007).
A. A. Govyadinov and V. A. Podolskiy, “Gain-assisted slow to superluminal group velocity manipulation in nanowaveguides,” Phys. Rev. Lett. 97(22), 223902 (2006). [CrossRef] [PubMed]
K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007). [CrossRef] [PubMed]
A. A. Oliner and T. Tamir, “Backward waves on Isotropic Plasma Slabs,” J. Appl. Phys. 33(1), 231–233 (1962). [CrossRef]
H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006). [CrossRef] [PubMed]
H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007). [CrossRef] [PubMed]
V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007). [CrossRef]
S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9(5), 407–412 (2010). [CrossRef] [PubMed]
G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]
E. Feigenbaum and M. Orenstein, “Modeling of Complementary (Void) Plasmon Waveguiding,” JLT 25, 2547–2562 (2007).
M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 (2004). [CrossRef] [PubMed]
H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]
E. Feigenbaum and M. Orenstein, “Ultrasmall volume plasmons, yet with complete retardation effects,” Phys. Rev. Lett. 101(16), 163902 (2008). [CrossRef] [PubMed]
C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001). [CrossRef] [PubMed]
E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative dispersion: a backward wave or fast light? Nanoplasmonic examples,” Opt. Express 17(21), 18934–18939 (2009). [CrossRef]
G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67(3), 035109 (2003). [CrossRef]
A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef]
M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87(25), 253902 (2001). [CrossRef] [PubMed]
T. Tamir and A. A. Oliner, “The Spectrum of Electromagnetic Waves Guided by a Plasma Layer,” Proc. IEEE 51(2), 317–332 (1963). [CrossRef]

J. Appl. Phys.

A. A. Oliner and T. Tamir, “Backward waves on Isotropic Plasma Slabs,” J. Appl. Phys. 33(1), 231–233 (1962). [CrossRef]

JLT

E. Feigenbaum and M. Orenstein, “Modeling of Complementary (Void) Plasmon Waveguiding,” JLT 25, 2547–2562 (2007).

Nat. Mater.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9(5), 407–412 (2010). [CrossRef] [PubMed]

Nat. Photonics

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007). [CrossRef]
M. Sandtke and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photonics 1(10), 573–576 (2007). [CrossRef]

Nature

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007). [CrossRef] [PubMed]
C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001). [CrossRef] [PubMed]

Opt. Express

E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative dispersion: a backward wave or fast light? Nanoplasmonic examples,” Opt. Express 17(21), 18934–18939 (2009). [CrossRef]

Opt. Lett.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef]

Phys. Rev.

R. H. Ritchie, “Plasma Losses by Fast Electrons in Thin Films,” Phys. Rev. 106(5), 874–881 (1957). [CrossRef]

Phys. Rev. B

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67(3), 035109 (2003). [CrossRef]

Phys. Rev. Lett.

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 (2004). [CrossRef] [PubMed]
H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]
E. Feigenbaum and M. Orenstein, “Ultrasmall volume plasmons, yet with complete retardation effects,” Phys. Rev. Lett. 101(16), 163902 (2008). [CrossRef] [PubMed]
A. A. Govyadinov and V. A. Podolskiy, “Gain-assisted slow to superluminal group velocity manipulation in nanowaveguides,” Phys. Rev. Lett. 97(22), 223902 (2006). [CrossRef] [PubMed]
H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006). [CrossRef] [PubMed]
M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87(25), 253902 (2001). [CrossRef] [PubMed]

Proc. IEEE

T. Tamir and A. A. Oliner, “The Spectrum of Electromagnetic Waves Guided by a Plasma Layer,” Proc. IEEE 51(2), 317–332 (1963). [CrossRef]

Science

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007). [CrossRef] [PubMed]
G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]

Other

M. Orenstein, “Slow Light by Slow Waves: Plasmonics for Light Halting,” Topical Meeting on Slow and Fast Light 2007, Salt-Lake City Utah, USA (2007).

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