The exotic regime of negative dispersion, defined here as a spectral range where the phase and group velocities are opposite in direction, is a subject of seemingly two disjointed research areas: one is the intriguing field of fast light [parallel phase and energy and antiparallel pulse (group) velocity] [1–4], and the other is the highly active field of negative-index metamaterials (parallel pulse group and energy and antiparallel phase velocity) [5–7]. In the community of negative index, a frequent distinction is that negative group velocity indicates negative refraction, and sometimes more cautiously that antiparallel group and phase velocity indicate negative index (see [8–10]). In this work we show that these assertions should be readdressed more rigorously since causal systems with negative dispersion can be attributed to either negative group—fast light or negative index backward wave—but not to both simultaneously. While it is always true that in negative-index structures the energy velocity is positive (from the source) and phase velocity is negative [7], energy velocity is not directly provided by the dispersion curves, which are derived many times from the stationary solutions of the system (e.g., waveguides) and do not include causality.

We show here that properly selecting the viable solution from the real dispersion curve may be accomplished by checking for causality by artificially incorporating a small loss in the dielectric constant; however, this necessitates additional data beyond the dispersion relations, namely the explicit dependence of the dispersion on the dielectric constant, which, in general, is not trivial or not known a priori. Alternatively, we reveal causality by exploring the physical mechanisms leading to the negative dispersion. Subsequent to the general arguments on negative dispersion, we exemplify the fast-light and negative-index modes of operation for a plasmonic waveguide, which is comprised of a thin dielectric slab between two thick metal layers (plasmonic gap waveguide) that is known to exhibit negative dispersion for both fundamental guided modes [11,12]. Two interesting reports, one theoretical [13] exploiting the TM1 mode of such a gap waveguide to design a perfect plasmonic lens and an experimental measurements of negative refraction [14] in a similar structure, are highly related to our discussion.

A segment with negative slope of the wave dispersion relations, shown schematically in Fig. 1(a) , is the starting point of our discussion. We are using the wave phase notation $\text{exp}\left\{\text{i}\omega \text{t}\mp i\beta \text{z}\right\}$ where ω is the angular frequency, β is the eigenwave vector (z direction) magnitude β = n _eff k ₀, n _eff is the modal effective index, and k ₀ the free-space wave vector magnitude. This dispersion curve, if taken literally as depicted, represents fast light and not negative index, since n_eff is positive and the group velocity (slope) is negative. However, for a reciprocal structure (as most structures in photonics are) one can extend the dispersion curve by its mirror image [dashed line in Fig. 1(a)]. The latter, as depicted, is evidently negative index dispersion, having a negative phase velocity and a positive group velocity (slope). The selection of the valid physical solution from the dispersion cannot be performed without further considerations; however, in any causal negative dispersion scenario only one of the two modes of operation will be causal, as will be shown in the following discussion.

 

Fig. 1 (a) Schematic presentation of a negative dispersion curve. Block blue arrows, group direction; block red arrows, phase direction. (b) Schematic roots in the complex plane of modal propagation constant for negative dispersion. Green points on the real axis indicate the solution pair for the lossless case, and the red/blue arrows indicate the revolution of the roots into the causal fast-light/backward-wave quadrants.

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The way causality discriminates between the two cases is illustrated in the complex refractive index plane of Fig. 1(b). For determining causality the wave source should be selected (e.g. at $z\to -\infty$in our case, launching power in the positive z direction). Only solutions with negative imaginary n_eff (lower half plane), i.e., having a decaying amplitude in the positive z direction, are causal solutions for passive material (and our source selection) [15]. The upper half plane is thus of no interest to us since the energy velocity there is negative. Quadrant (III) exhibits causal backward waves (negative index), while quadrant (IV) exhibits causal fast light solutions. When loss is added rigorously to the structure analysis, the pair of real coexistent roots will revolve into the complex plane—either clockwise where fast light is the causal solution or counterclockwise where negative index is the chosen solution. Therefore, for any negative dispersion scenario, both phenomena cannot be exhibited simultaneously.

As hinted before, inaccurate assertions contributing somewhat to the ongoing confusion between the fast light and negative index may stem from attempts to extract information from incomplete dispersion relations. A prominent fault is extracting information from the first quadrant (positive β) of a dispersion curve that is derived by electromagnetic modal calculations and is the most common way dispersion relations of structures (e.g., waveguides) are presented in the professional literature [16,17]. To illustrate this point we present in Fig. 2(a) the first quadrant of the calculated dispersion for a plane wave propagating in a bulk medium having ε = μ (dielectric constant = magnetic permeability), each characterized by a single (causal) Lorentian dispersion. The real parts of the index of refraction n includes 3 regimes of positive dispersion (I, III, V) and 2 regimes of negative dispersion (II, IV), which may look without any further information as a manifestation of 3 lossy photonic bands, with II and IV as bandgaps. Although this dispersion curve is the correct stationary solution of the structure, it does not discriminate between solutions excitable by sources at + or – infinity. When selecting the energy propagation direction (e.g., to + z), it is obvious that the above interpretation is completely wrong, as is evident from the resulting causal dispersion curve of Fig. 2(b) —in II, the negative dispersion is a fast light, in IV it is negative index, and in III the seemingly positive dispersion is rather an exotic combination of fast light and negative index, similar to the recently measured light in a double-negative metamaterial [4].

 

Fig. 2 Plane wave dispersion for a bulk material having equal dielectric constant and magnetic permeability, each with a Lorentian dispersion. (a) Stationary solutions of the real effective index (n_r) presented in the positive quadrant (energy propagation direction is not determined). (b) Causal solution for a source located at -∞ [imaginary part of the effective index (n_i) overlay by dotted line; the resonance frequency is at the peak of n_i(ω)].

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We proceed by examining an important practical system, the plasmonic gap structure of Fig. 3(a) , which supports only the TM0 and TM1 propagating modes for nanometric thin gaps. For the (fictitious) lossless case [Fig. 3(b)] it is perceived [11] that the TM1 mode can exhibit a negative dispersion slope at very thin gap and at frequencies near or above the surface plasmon polariton resonance. The textbook dispersion curves of such a waveguide (e.g., in [16]) are depicted for positive n _eff values only and are due to the reciprocity we may extend them symmetrically to negative n _eff values, as shown in Fig. 3(b) for the gold–Si–gold gap. This dispersion curve presumably indicates fast light [RHS of Fig. 3(b)] or backward waves [LHS of Fig. 3(b)].

 

Fig. 3 (a) A plasmonic-gap waveguide scheme with magnetic-field amplitude of TM0 and TM1 modes. (b) Non-causal dispersion curves of (lossless) gold–silicon–gold plasmonic gap for TM0 and TM1 at different gap widths. Textbook dispersion is curve shown in the first quadrant (positive n _eff) marked by the bold blue frame. (c) Poynting vector in the propagation direction (z) for the negative index TM1. (d) Causal normalized dispersion curve for energy propagating to + z. Stars (yellow, green) for the (30nm, 20nm) gaps indicating the cutoff transition (zero group velocity) between forward and backward waves.

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One can select the viable physical solution without adding loss by examining the specific mechanisms determining the phase and energy flow in the system. From first principles, the power in the dielectric gap (ε > 0) propagates always (for lossless cases) in the same direction as the phase, whereas the power in the metallic cladding (ε<0) propagates against the phase direction. Therefore, if the predominant part of the power propagates in the dielectric material, it has a positive real part and vice versa. The transition between the negative and positive index can be identified as the minimum point of group velocity = 0 or stopped light [stars in Fig. 3(d)], similar to what was analyzed for waveguides comprised of a negative index medium (e.g., [18]). The transverse spatial distribution of the z component of the time-average Poynting vector is plotted in Fig. 3(c) at a specific frequency (ω/ωp = 0.28, d = 30nm, and n _eff = −3.86) for which TM1 exhibits negative index. The total power (by integration) flows in the positive z direction, indicating phase and energy velocities in opposite directions, validating that the mode is clearly the negative-index mode with positive dispersion slope. The proper causal dispersion curves of the gap plasmon are therefore those depicted in Fig. 3(d). It should be re-emphasized here that the textbook dispersion curves, e.g., the positive n _eff part of Fig. 3(b), may be portraying an erroneous picture—e.g., at a gap width of 30nm, mode TM0 and mode TM1 with negative group velocity are both forward-propagating modes and the dispersion curve of TM1 is continuous, while actually it exhibits a cutoff with a prominent branch point at the zero slope value [stars in Fig. 3(d)].

When actual metal losses are included, a significant distortion of the real effective index occurs. Both propagating modes (TM0 and TM1) now exhibit regions of negative dispersion. While in the lossless case the negative dispersion stems only from backward waves, when the actual loss is presented a fast light solution may emerge as well.

The dispersion of the lossy TM1 mode is depicted in Fig. 4(a) for a thin gap (d = 30nm) (complex permittivity values of gold are taken from measurements [19]). We examine solutions having Im{n _eff)<0, especially those marked by a red ellipse, which are well- propagating modes having a modal figure of merit FOM = Re{n _eff}/Im{n _eff}>1. The TM1 mode is propagating near and below wavelength λ<630 nm with a negative real effective index and positive group velocity, a clear case of a negative index and a forward-propagating group with a significantly slow velocity—Vg~0.05c (c is the light speed in vacuum). For completion we examine also the dispersion curve section marked by the green ellipse in Fig. 4(a). Here the dispersion is positive—phase and group velocities are parallel; however, both of them are antiparallel to the energy flow direction. Thus, both phase and group velocities are negative, resulting in a fast-light negative-index wave. Such weird solutions exist primarily within the bandgap of the related resonance (characterized by FOM<1) and thus is always accompanied by a very limited propagation length. However, it reemphasizes that the usual requirement of antiparallel group and phase velocities for a negative effective index is not essential.

 

Fig. 4 Causal complex normalized dispersion curves for actual gold–silicon–gold plasmonic gap with gap thickness d = 30nm. Red, green ellipses indicating regions of interest, as discussed in the text. (a) TM1, (b) TM0.

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The dispersion curve for the TM0 mode is shown in Fig. 4(b). It is apparent that the mode is a well-propagating mode for λ>660 nm, and it exhibits a negative dispersion over a narrow frequency range around λ~680nm (red ellipse). Here the mode has a positive effective index and a negative group velocity—a fast light solution that does not exhibit negative refraction but rather backward propagation of a pulse envelope (at a slow velocity of Vg~-0.01c).

The analysis for the lossless case has indicated that the backward propagation of the wave stems from the counter propagation of the electromagnetic power in the dielectric and in the metal. Even though it is not always the case when loss is introduced, in the regions of interest (FOM>1) it is still valid. For the TM1 mode at the negative dispersion regime, more power propagates in the metal and thus the overall power counter propagates the phase [Fig. 5(a) ]. The slow light characteristics also originate from the fact that a significant part of the power is propagating backward, diminishing the forward power flow. The energy velocity for the propagating mode, $V_e^{}=\genfrac{}{}{0.1ex}{}{S_z}{U}=\genfrac{}{}{0.1ex}{}{\colorbox[rgb]{}{$\genfrac{}{}{0.1ex}{}{1}{2}$}\mathrm{Re}\left\{{\displaystyle \colorbox[rgb]{}{$\iint \left(E\times H^{*}\right)\cdot \widehat{z}$}}\right\}}{\colorbox[rgb]{}{$\genfrac{}{}{0.1ex}{}{1}{4}$}{\epsilon }_0{\displaystyle \colorbox[rgb]{}{$\iint {\partial }_{\omega }\left(\omega {\epsilon }^{\prime }\right){\left|E\right|}^2+{\eta }_0^2{\left|H\right|}^2$}}}$, is equal to the group velocity in this case, as is evident in Fig. 5(b), even though the loss is not negligible (U is the electromagnetic stored energy, E and H are the electric and magnetic fields, and ε₀ and η₀ are the vacuum permittivity and wave impedance, respectively).

 

Fig. 5 Energy velocity and power flow ratios for TM1 and TM0 modes of the plasmonic gap waveguide in Fig. 4. Red ellipses are related to those of Fig. 4; (a) TM1 and (c) TM0 ratio of total power (z direction) in the dielectric to that in the metallic cladding, (b) TM1 and (d) TM0 ratio of energy velocity to group velocity.

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The TM0 mode in the fast-light regime (negative group velocity) has more power propagating in the dielectric than in the metal, thus the phase and overall power are propagating in the same direction [see Fig. 5(c)], and the energy propagation velocity is not only opposing to the group velocity direction, their magnitudes are not related [Fig. 5(d)] due to the highly dispersive large imaginary part of n _eff.

We showed that what is perceived as negative dispersion cannot be attributed to either negative index or fast light before applying causality considerations, either formally or by a detailed understanding of the material and structural source of the negative dispersion. Details for a plasmonic waveguide structure confirm both negative index and fast light (and even a fast light/negative index wave with positive dispersion), but for different modes and at different wavelengths. A detailed understanding of the dispersion mechanisms is important for correct determination of the mode of operation. The plasmonic gap structure is an excellent structure to realize this variety of anomalous dispersion phenomena in two dimensions—e.g., the measurements of negative refraction in a gap structure [14], which is a step toward visible light 2D negative-index schemes—which is highly important for the metamaterials community.