Plasmonics provides a unique platform for controlling electromagnetic propagation due to the electric field confinement and enhancement it enables, and reveals exciting possibilities for next-generation photonic devices [1]. While much literature regarding plasmons excited at visible frequencies can be found, terahertz (THz) plasmons remain relatively unexplored. Nonetheless, it has been shown that THz electromagnetic energy can be transported across a dense collection of metallic particles via the near-field coupling of non-resonant particle plasmon oscillations [2,3]. By exploiting the parameters of such transport unique photonic behaviour can be revealed, such as magnetic dependent transmission mediated by magnetoresistance in the metallic particles [4,5] and electron-spin dependent electromagnetic transport [6]. Here, we demonstrate a new terahertz plasmonic composite that is random in nature and exhibits an atypical refractive index. We show that the group refractive index can be increased to be larger than the effective refractive indices of the constituent metallic and dielectric parent composites. The behaviour of the composite relies on interplay between two different photonic transport regimes which, to our knowledge, has not previously been theoretically or experimentally discussed.

In describing the permittivity and the permeability of a homogenized composite random media, Burgmann’s effective medium theory (EMT) is widely utilized [7-9]. Within the theory’s framework, the electromagnetic response of a particulate composite medium is described by a set of average constitutive parameters, such as effective permittivity (ε_eff) and permeability (μ_eff), representing a weighted average of the constitutive. However, this homogenization formalism is valid only for ensembles composed of subwavelength-scale dielectric media and/or nano-scale metallic media, where the size of the metallic inclusions, a, is less than the radiation skin depth, α [10]. It is important to realize, however, that for random composites comprising metallic elements where a ≫ α, EMT does not capture the essence of the electromagnetic interaction. In this regime, a formalism describing mutual particle plasmon interactions within and between metallic elements is required [2,3].

The random composite employed in this work consists of a 4 mm thick mixture of Co metallic microparticles having a mean dimension of a = 74 ± 25 μm and spherical sapphire dielectric microparticles having a = 100 ± 15 μm [Fig. 1(a)]. The intrinsic relative permittivities of the Co and sapphire particles at 1 THz are -8×10³ + 10⁴ i [11] and 10.5 + 0.02i [12], respectively. We conduct our experimental observation in the terahertz (THz) regime where we access the time-domain electric field waveforms and their polarization states. Since α for Co at 1 THz is ~ 100 nm [11], which is much less than a, the ε_eff of the composite metal-dielectric random medium cannot be described as an effective medium. The electromagnetic behaviour of ε_eff is depicted in the group refractive index or the group velocity of a THz pulse propagating through the composite. By incorporating an increasing metallic particles volume fraction, f, into the dielectric random particles ensemble, we show evidence of a transition where there is a marked range of f where the refractive index is higher than the independent refractive indices of metallic and dielectric particle ensembles.

 

Fig. 1. (a) Schematic diagram illustrating the random composite consisting of polydispersed metallic and dielectric particles. The circled images depict scanning electron microscope images of the dielectric and metal particles and an optical image of the composite. (b) Time-domain terahertz waveforms transmitted through an empty sample cell (reference) and 4.0 mm thick Co/sapphire particle mixtures for Co particles volume fraction, f, varying from 0.0 % to 100 %.

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The composite is excited with single-cycle THz pulses derived from a 100 μm gap, semi-insulated GaAs photoconductive emitter illuminated with 20 fs, 800 nm laser pulses. The on-axis electric field transmission through the sample is measured via electro-optic sampling in a <111> ZnSe crystal [13]. Shown in Fig. 1(b) are representative time-domain THz electric field waveforms transmitted through the composite where f is increased from 0.0 % (dielectric particles) to 100 % (metal particles). It is important to note that for representative particle ensembles with f= 0%,f= 17%,f= 27%, and f= 100%, the packing fraction (ρ) is measured to be 0.60 ± 0.01, 0.61 ± 0.01, 0.58 ± 0.01, and 0.58 ± 0.02, respectively. From these values, it is clear that the ρ remains fairly consistent over the full range of f. The electric fields shown in the figure are polarized parallel to the incident polarization. The most notable feature is that as f is increased greater than 17 %, an appreciable advancement of the pulse arrival time is observed. This extraordinary behaviour is quantified by assigning the pulse arrival time as the temporal position of the peak pulse intensity (corresponding to the time when the greatest number of photons are detected), T [Fig. 2(a)]. Figure 2(b) plots the time-dependent intensity profiles of the transmission electric field components polarized parallel, E _∥, and perpendicular, E _⊥, to the incident polarization for various f. For f = 0.6%, the electric field is polarized along the incident polarization, since the majority of the pulse energy is in the parallel component. The small perpendicularly-polarized field component is attributed to small angle scattering from the dielectric constituents. When f increases to 40 %, T is delayed by 1 ps. This delay is a result of the THz pulse propagating a longer optical path due to increased scattering from both the dielectric and metallic particles, and the equal intensity of the parallel and perpendicular field components indicates that the transmitted electromagnetic pulse is unpolarized. Such polarization-randomizing scattering events impair the coherence of the electromagnetic energy transport. However, at f = 100 % the transmitted pulse arrives 2 ps earlier than the transmitted pulses measured at f = 0.6 %. The high polarization purity of the THz electric field suggests coherent electromagnetic energy propagation, and such marked transmission through the metallic Co particles is attributed to the recently demonstrated plasmonic enhanced propagation [2,3]. Effectively, the addition of opaque, metallic particles to the composite in this regime reduces the optical path length.

To shed light on the frequency dependence of this phenomenon, the transmission power spectra is explored as shown in Fig. 2(c). The main features of this data are the frequency-dependent attenuation for f < 17.0% and the frequency-independent attenuation for f > 17.0%. The transmission through the dielectric ensemble (f = 0%) encompasses frequency components from 0.1 THz to 0.8 THz. However, as f increases to 7.2%, nearly all the frequency components above 0.4 THz are extinguished. Preferential attenuation of the higher frequency components is ascribed to the frequency-dependent Rayleigh scattering from the metallic particles [14], where shorter wavelength radiation is more readily scattered from the on-axis direction relative to larger wavelength radiation. It should be noted that scattering events from the individual metallic particles cannot account for the frequency-independent attenuation that is observed for f between 17.0 % and 72.6 %.

To illustrate the observed behaviour of the composite, we extract the effective refractive index, n_eff, as a function of f. Figure 3(a) displays the change in the effective refractive index, ∆n_eff, (measured relative to that of a sample having f = 0%) versus f over the frequency range of interest. Within the 0.1- 0.3 THz bandwidth, the refractive index can be significantly altered by the addition of the metallic particles; and more interestingly, n_eff can be made higher than the refractive indices of parent dielectric (n_eff^d) or metallic (n_eff^m) constituents. This behaviour is further illustrated in Fig. 3(b) for a representative n_eff versus f plot at a frequency of 0.23 THz. Within the range 0< f < 20%, n_eff is shown to increase from 1.81 to 1.91 (∆n_eff >0); however, beyond f = 20%, n_eff displays a marked decrease, yet the refractive index is higher (ie: ∆n_eff > 0) than either n_eff^d = 1.81 or n_eff^m = 1.66 (Note that the intrinsic index of individual opaque Co particles and transparent sapphire particles at 0.23 THz are 170+200i [11] and 3.25 [12], respectively). Surprisingly, despite the high volume fraction of opaque metallic particles, at f = 70% ∆n_eff = 0. That is, the dielectric-metal random composite has the same refractive index (n_eff^d = 1.81) as a dielectric particle ensemble. Beyond f = 70%, ∆n_eff< 0 and approaches a value of ∆n_eff = -0.15 at f = 100%. Notably, for a metallic particle ensemble (f = 100%), n_eff^m is 8.3% lower than that of a pure dielectric particle ensemble (n_eff^d).

 

Fig. 2. (a) Illustrative diagram depicting the THz electric field (E_THz) transmitted through the random metallic-dielectric composite with components polarized parallel (E _∥) and perpendicular (E _⊥) to the incident polarization and temporal position shifted by a time T relative to the transmission through a dielectric particle sample (upper left). (b) Time-dependent THz pulse intensity envelopes through a 4.0 mm thick sample having various f values for both parallel and perpendicular polarization states. Here, the time axis is measured relative to a 4.0 mm thick pure dielectric ensemble. (c) Power spectra of a THz pulse transmitted through mixtures of metallic and dielectric particles for varying f. The band marks the frequency range which is shared among all the samples of different f.

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Fig. 3. (a) Plot of the change (relative to a dielectric ensemble) of the effective refractive index of the random composite as a function of f and frequency. (b) Effective refractive index of the random metallic-dielectric composite measured at 0.23 THz and various f. Notably, within the atypic range the refractive index is larger than either the dielectric (n_eff^d) or metallic (n_eff^m) ensembles refractive indices.

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It is evident that the group refractive index is strongly dependent on the effective optical path which is related to f, or alternatively, the metallic particle separation. The average inter-particle separation, S_ave, between metallic particles is estimated by approximating the metallic particles as isotropically distributed random spheres, as shown in the inset of Fig. 4. The average radius, r_ave, occupied by a single metallic particle in a volume, V_ave, is r_ave = (3V_ave/4π)^1/3, and thus the average physical separation between nearest neighbour metallic particles from the surface of the particles is save = 2r_ave - a. Since the addition of metallic particles displaces a volume otherwise occupied by dielectric particles, the effective optical inter-particle separation, S_ave,opt = s_ave n_s, must account for the weighted refractive index, ns, of the dielectric particles surrounding the metallic particles. Here, n_s is calculated using a Beer effective medium description [10],

 

Fig. 4. Group velocity (normalized to c) versus the average optical separation between metallic particles. The inset illustrates the average optical separation of metallic particles immersed in a background of a dielectric particle ensemble, where r_ave is the average radius occupied by a single metallic particle in a volume V_ave and a is the average diameter of a metallic particle. The solid line in the figure is not a theoretical curve but a fitted curve.

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where n_d ≈ 3 is the refractive index of sapphire at 1 THz [12], V_t is the total volume occupied by the composite, and V_d is the total volume of sapphire in the composite. It is instructive to analyze the group velocity, v_g, of the light pulse versus S_ave,opt. As depicted in Fig. 4, for S_ave,opt > 288 μm, v_g is relatively constant at ~ 0.54c. In this regime, the metallic particles are separated by distances greater than the diameter of the dielectric particles and, on average, the spacing between metallic particles is occupied by several dielectric particles. Thus, mutual plasmonic coupling between metallic particles is inhibited and the metallic particles act like independent scatterers removing electromagnetic energy from the incident beam via Rayleigh scattering. Since the transmission in this regime is highly polarized, the pulse transmission through the composite arises mainly from direct propagation through the transparent dielectric particles (ballistic transmission) and small angle scattering. At S_ave,opt ~ 170 μm, v_g reaches a minimum of 0.51c, which is lower than the group velocity in either constituent. The reduction in v_g over the range 170 μm < S_ave,opt < 288 μm is attributed to augmented scattering by the metallic and dielectric particles. Here, the ballistic transmission efficiency diminishes, and a greater portion of the transmitted pulse is mediated by scattered electromagnetic waves that must propagate within the gaps between the opaque metallic particles. Scattered photons following such a “zig-zag” path thus results in an increased propagation path-length and, accordingly, a decreased v_g and increased n _eff. However, for S_ave,opt < 170 μm v_g begins to increase and approaches 0.58c at S_ave,opt ~ 14 μm. It is noteworthy that the inflection in v_g at S_ave,opt ~ 170 μm corresponds to a physical separation of S_ave ~ 95 μm, similar to the diameter of one dielectric particle (100 ± 15 μm). At this inflection value of S_ave,opt, the metallic particles become close enough to each other, on average, such that near-field particle-plasmon interaction between metallic particles begins to govern the propagation of electromagnetic energy across the ensemble [2,3]. As such, the increasing v_g as S_ave,opt decreases from 170 μm to 14 μm is attributed to increased particle-plasmon coupling. Note that the narrow band of the transmitted pulses for S_ave,opt < 170 μm (i.e. f > 17%) supports this claim, since near-field particle-plasmon transport is supported only over a relatively narrow band [2,3]. As the average spacing between metallic particles becomes significantly less than the diameter of the dielectric particles, the absence of straight-line trajectories through the dielectric particles precludes ballistic electromagnetic pulse propagation. Accordingly, nearly all transmission is mediated by near-field particle-plasmon interaction between metallic particles for S_ave,opt near 14 μm. Note that we can also identify the inflection point at save,opt ~ 170 μm as the particle plasmon transport percolation threshold, separating regimes where the electromagnetic behaviour in the composite is governed by dielectric scattering and plasmonic interaction. At the plasmonic percolation threshold, the material exhibits atypical photonic transport that is mediated by contributions from both ballistic scattering and near field plasmonic coupling. To our knowledge, there are no existing theoretical models to describe this behaviour.

In this work we explored a new non-resonant composite consisting of dense ensembles of subwavelength-sized dielectric and metallic constituents. Using time-domain THz spectroscopy, the random composite is shown to exhibit atypical group refractive index. We experimentally verified that the group refractive index can be increased to a larger value than the refractive indices of the constituent parent composites. This peculiar behaviour cannot be described by conventional effective medium theory, but can be qualitatively explained by near-field particle-plasmon coupling between the metallic particles and scattering. However, a more rigorous theoretical treatment is still required. This work introduces a new platform for material design where random structural configurations preclude the necessity for complex fabrication, and opens the door for engineered photonic materials having customizable and/or atypical optical constants.