A class of active terahertz devices that operate via particle plasmon oscillations is introduced for ensembles consisting of ferromagnetic and dielectric micro-particles. By utilizing an interplay between spin-orbit interaction manifesting as anisotropic magnetoresistance and the optical distance between ferromagnetic particles, a multifaceted paradigm for device design is demonstrated. Here, the phase accumulation of terahertz radiation across the device is actively modulated via the application of an external magnetic field. An active plasmonic directional router and an active plasmonic cylindrical lens are theoretically explored using both an empirical approach and finite-difference time-domain calculations. These findings are experimentally supported.
© 2009 Optical Society of America
The nature of metallic-dielectric interfaces and their ability to support collective and coherent electron oscillations, known as plasmons, has recently garnered significant attention. While much research has dealt with resonant plasmons at optical frequencies , non-resonant plasmons at terahertz (THz) frequencies have received considerably less attention. This spectral regime provides an attractive alternative to optical frequencies due to low losses in metals at THz frequencies resulting from weak confinement of the radiation to the metallic surface. As such, THz plasmonics may be influential in the realization of practical applications including ultrahigh bandwidth interconnects and low-loss wave guiding. For instance, the ability of THz pulses to be guided by a metallic wire  and periodically perforated films  show promise for plasmonic-enabled information transmission at ultrahigh bandwidths. In contrast to THz engineered structures, the ability of THz particle plasmons to near-field couple and coherently transport THz radiation over large distances introduces a new paradigm for actively controlling the propagation of radiation [4,5]. While the realization of active THz devices has been limited, such advances could allow a much wider range of THz device functionality. Interestingly, particle plasmon-enhanced THz propagation introduces new degrees of freedom that permit interesting phenomena including photonic anisotropic magnetoresistance [6,7], spinplasmonic photonic control , and materials exhibiting an atypical effective refractive index that cannot be explained by effective medium theory .
In this work we present a class of particle plasmon enabled THz propagation where the position-dependent phase accumulation of THz radiation can be actively tuned via the application of an external magnetic field. Here, we exploit the local properties of particle plasmon coupling, where the phase accumulation is sensitive to both the intrinsic properties (e.g. electron-spin dependent permittivity) and extrinsic properties (e.g. optical separation) of the metallic particles. As such, in a particle composite containing mixtures of subwavelength cobalt (Co) and sapphire particles, we target the intrinsic properties via spin-orbit interaction manifesting as anisotropic magnetoresistance (AMR) in the Co particles, and we target the extrinsic properties via a spatial dependence of the concentration of sapphire particles. By implementing various spatial dependencies of the sapphire particle concentration, a new class of actively controlled THz plasmonic devices having unique modes of operation, such as directional routers and active lenses, is introduced. Preliminary experimental results for deflection can be found elsewhere  and, for completeness, we include the primary findings here. We anticipate the maximum possible deflection for a directional router and minimum focal length for a lens using this approach to be 20° and 3.2 mm, respectively. Considering that the realization of active control of light is paramount for the development of the next generation of THz photonics based information technologies, this work introduces a platform for active plasmonic THz devices.
2. Model description
To describe the phenomenon, THz transmission through a random ensemble of subwavelength sized Co and sapphire particles is considered. The relative occupancy of the two species is characterized by defining the volume fraction of Co to sapphire,
where V C and V S are the total volumes of the Co and sapphire particles, respectively. When τ C < 0.67, the THz radiation transport through the random ensemble is primarily mediated by ballistic small-angle forward scattering of photons through the sapphire particles; however, when τ C > 0.67 transport is mediated by the near-field radiation coupling via Co particle plasmons . In this regime, the effective refractive index, n, has been shown to linearly decrease with increasing τ C (i.e. n ∝ -τ c), where the removal of the sapphire particles results in a decrease of the effective optical distance between the Co particles which thereby enhances the near-field plasmonic coupling. Conversely to n, the effective absorption was relatively constant for all τ C > 0.67 .
We first consider a directional router that actively controls the direction of propagation of THz radiation, where we specify that τ C varies linearly from 0.67 to 1 across the transverse spatial extent of a particle ensemble, as shown in Fig. 1(a). As such, a device having a linearly graded effective refractive index is realized, and THz radiation transmitted through it exits with a tilted wavefront. The expected deflection angle, Θ, can be calculated by considering the optical path length of two rays sharing a common wavefront that pass through the τ C = 0.67 and τ C = 1 regions of the ensemble having indices of refraction n 0.67 and n 1, respectively. As shown in Fig. 1(a), since n1 < n 0.67, the ray that passes through the τ C = 1 region travels an extra distance, a = wsinΘ, where w is the total width of the gradient section of the sample. Moreover, for the two rays to share a common wavefront upon exiting the ensemble of thickness d, their optical path lengths must be equal and, accordingly,
Note that Eq. (2) is satisfied whenever (n 0.67 - n 1) << (n 0.67 + n 1)/2, which allows the slight curvature of the rays in the sample to be ignored. To account for deviations from perfect linearity of the effective index, a figure of merit, 0 ≤ Q 1 ≤1, is defined and, as such, one can rearrange Eq. (2) to obtain
To actively control the phase accumulation of the THz radiation and, accordingly, actively tune Θ, the AMR inherent to the Co particles is utilized. In ferromagnetic metals, the resistivity, ρ, depends on the application of an external magnetic field, B, through the relation ρ(B) = ρ0 + ρ Lorentz(B) + ρ AMR(B), where ρ 0 is the background resistivity, ρ Lorentz(B) is the contribution from Lorentz force that is present in all metals, and ρ AMR(B) is the contribution from AMR that is only present in ferromagnetic metals due to spin-orbit interaction . Interestingly, when THz propagation is mediated by particle plasmonic oscillations, it has been shown that AMR results in a delay, Δ τB, of the arrival times of THz pulses transmitted through dense ensembles of ferromagnetic microparticles . The degree of delay was shown to scale with the strength and direction of the magnetic field and the sample thickness. Notably, ΔτB is maximum for B aligned parallel to the THz electric field polarization direction, which will be the orientation assumed in the proceeding discussion. The thickness dependence was demonstrated to be linear, indicating that ΔτB is a cumulative effect depending on the number of Co particles traversed by the radiation. Therefore, ΔτB can be expressed as
where N C is the number of Co particles in the sample and K 1(B) is a B-dependent delay per unit of linear Co particle density perpendicular to the THz wavefront. For the ensemble of Co and sapphire particles investigated here, Eq. (1) can be expressed as τ C = (〈VC〉Nc)/(V Cell pf), where 〈Vc〉 is the average volume of a single Co particle, VCell is the volume of the sample cell, and pf = (V C + V S)/V Cell is the total particle packing fraction for the ensemble. Note that here we assume pf remains constant for all τ C if the sapphire and Co particles are chosen to be of similar size. Accordingly, N C in Eq. (4) can be replaced with the above to obtain
As such, ΔτB coincides with an increase in the effective refractive index, ΔnB, given by ΔnB = (cΔτB)/d, where c is the speed of light in a vacuum. Notably, since both ΔnB ∝ τ c and n ∝ -τc, the functional form of the spatial dependence of the overall effective index of refraction is preserved with the application of B for any arbitrary spatial variation of τ C. Accordingly, the overall effective index of refraction of the ensemble studied here maintains its linear gradient with the application of B. Nevertheless, the corresponding B-dependent angular deflection is given by
where K 2(B) = (cK 1(B)VCell pf)/(〈VC〉d 2) is a dimensionless parameter that results in a reduction of Θ as B is increased, as depicted in Fig. 1(b). K 2(B) is an intrinsic parameter that is not restricted to the above geometry but, rather, describes the net phase accumulation modulation resulting from the application of B for any device implementation that employs a spatial gradient of τ c . For the directional router, at large values of B (i.e. large K 2) it would be possible for the deflection to change to the opposite direction. Therefore, the application of B allows the active bi-directional control of the propagation of THz radiation.
It is interesting to note that behaviour other than directional routing could be realized by using more sophisticated geometries. Accordingly, we consider an ensemble that cylindrically focuses or defocuses the transmitted THz radiation by specifying a non-linearly varying τ C, such that n is maximum at the center of the device. With the aid of Fig. 2(a) and using a similar geometrical optics approach described above, it is seen that the ray that travels through the τ C = 1 region of the device passes through an extra distance, through air compared to the ray that travels through the τ C = 0.67 region, where F is the focal length. Therefore, F and the corresponding spatial profile of the index of refraction, n(x), are given by the following coupled equations:
where x is the transverse distance from the center of the plasmonic lens (Fig. 2(a)). Here, 0 ≤ Q 2 ≤ 1 is a figure of merit defined to account for imperfections of the effective refractive index spatial profile. For a typical lens where F > w, the focal length and refractive index are approximated to have the form
By implementing Eq. (5), it can be formulated that the application of B results in a magnetically dependent focal length given by
Interestingly, as K 2 increases and the denominator of Eq. (11) approaches 0, F approaches infinite values (i.e. collimated propagation), as shown in Fig. 2(b). Increasing K 2 to larger values results in a negative F that approaches 0 (i.e. diverging propagation).
3. Finite-Difference Time-Domain Calculations
To further explore the physical underpinnings of this phenomenon, numerical finite-difference time-domain (FDTD) calculations  are performed for a 3.1 mm thick collection of Co and sapphire particles. The diameters of the Co and sapphire particles are δ C = 75 μm and δ S = 100 μm, respectively, and the relative permittivity of the sapphire particles is set to γ = 10.47 . For the Co particles, a frequency-dependent Drude response, given by γ(ω) = 1-ωp 2/(iωτω + ω 2), is implemented using a plasma frequency ω p = 9.60 × 1014 s-1 and damping frequency ωτ = 8.85 × 1012 s-1 . In this study, two different sample geometries corresponding to a plasmonic router and a plasmonic lens, respectively, are explored. The particles are placed using pseudo-random number generation to determine the x- and y-coordinates, where the x- and y- directions correspond to the THz electric field polarization and THz pulse propagation directions, respectively. For the plasmonic router configuration, τ C is set to vary linearly from 1 to 0.67 over a distance of 3 mm by using a linearly sloped probability distribution for the randomly generated x-coordinates and a uniform probability distribution for the y-coordinates (Fig. 3(a)). The radiation source implemented is a single-cycle, 2 ps long THz pulse having a linear polarization along the x-direction. To ensure that the incident THz pulse spot does not extend beyond the gradient portion of the sample, a THz source having an intensity full-width-half-maximum (FWHM) of 1.1 mm is placed 0.26 mm in front of the device. Shown in Figs. 3(b)-3(e) is the time evolution of the transverse THz electric field, E, in 5 ps intervals as the pulse passes through the device. It is clear from these images that the wavefront gradually tilts toward the right side of the sample, where τ C is lowest. By viewing a snapshot of E at a time 50 ps after the THz pulse is incident on the device, at which point the pulse has propagated to the far-field, it is evident that the propagation direction is deflected towards the side of the sample with lower τ C, as shown in Fig. 3(f). Note that the divergence of the wavefront as the transmitted pulse propagates away from the sample is attributed to the relatively small transverse size of the incident THz radiation. By calculating the center of energy of the angular-dependent average pulse intensity at a time of 50 ps, an angular deflection from the y-axis of ΘFDTD = 8.4° is measured.
To study the behaviour of the plasmonic lens, τ C is varied to resemble a parabolic profile, as shown in Fig. 4(a). Here, a collimated THz pulse having an intensity FWHM of 3.2 mm is incident on the device. Shown in Fig. 4(b) is E at a time 40 ps after the THz pulse is incident on the device and, for comparison, the same pulse that has propagated through free space is shown in Fig. 4(c). It is apparent from these results that the transverse profile of E for the pulse transmitted through the device is narrower than that for the pulse transmitted through free space. A comparison of the transverse profiles of the normalized pulse intensities taken at the location, y 0, of maximum intensity, I(x) ∝ E(x, y 0)E *(x, y 0), is shown in Fig. 4(d). Here, the FWHM of I(x) can be clearly seen to decrease from 3.2 mm to 0.9 mm after the passage of the collimated pulse through the plasmonic lens. It is evident that the peak position has deviated from the center by 0.4 mm, which is attributed to random asymmetry of the concentration of sapphire particles due to the use of pseudo-random number generation to determine the particle positions. While this suggests that care must be taken to maintain symmetry when fabricating the device, it is expected that a physical implementation would be less sensitive to non-symmetries. That is, in the two-dimensional FDTD approach, the THz electric field interacts with fewer particles due to the lack of the third dimension. As such, the FDTD approach is more sensitive to non-symmetries due to an insufficient number of particles present to average out local fluctuations in τ C.
4. Experimental results
We investigate the THz transmission through a 3.1 mm thick ensemble of Co and sapphire particles having mean sizes of δ C = 74 ± 23 μm and δ S = 100 ± 12 μm, respectively. Linearly polarized, 2 ps long single-cycle THz pulses with a center frequency of 0.38 THz and spectral FWHM of 0.32 THz are generated via excitation of a GaAs photoconductive switch with 20 fs, 800 nm pulses from a Ti:Sapphire laser . As shown in Fig. 5(a), the THz pulses are incident on a device consisting of five 600 μm thick homogeneous regions having τ C = 0.67, 0.73, 0.81, 0.92, and 1, approximating a linearly graded τ C. It should be noted that the thickness of the regions is chosen to be less than the wavelength of the THz radiation to reduce the effect of the finite step-like variations in τ C. Since δ C ≈ δ S, the packing fraction (pf) of all regions is measured to be 0.59±0.1 which was a necessary assumption for the development of Eq. (5). To characterize the deflection induced by the device, an angularly ranging THz time-domain spectroscopy system is utilized, where temporally resolved THz pulses can be measured at any angle, θ, around the sample . The THz electric field and its polarization state are measured via electro-optic sampling in a <111> ZnSe crystal.
Accordingly, E for on-axis transmission (i.e. θ = 0°) through an empty sample cell, and E for the THz pulses transmitted through the device at θ = 20° to -5° are depicted in Fig. 5(b) and 5(c), respectively. From these results it is apparent that the THz electric field amplitude is maximum near θ = 5°. Additionally, it is evident that the leading lobes of the THz pulses are centered closer to 0° than the trailing lobes, which was also observed in the FDTD calculations. To quantitatively characterize the propagation direction of the transmitted pulses, the average energy flux,
is first calculated, where f is frequency, and <…>f denotes averaging over the signal bandwidth. As can be seen in Fig. 6(a), for the THz pulses transmitted through the device, an angular deflection with respect to pulses transmitted through an empty cell is observed. By identifying the deflection angle, Θ, as the center location of the Gaussian regression lines, a deflection angle of Θ = 5.7° is observed. To verify that the deflection is a result of the linearly varying τ C, the THz transmission through a 3.5 mm thick sample having τ C = 1 (S1) and a separate 3.5 mm thick sample having τ C = 0.67 (S2) is measured (Fig. 6(b)). As expected, for these two samples there is no appreciable deflection. Note that the THz pulses transmitted through S1 and S2 can also be used to determine n0.67-n1 (Fig. 6(c)). Accordingly, the 2 ps relative delay between the two pulses measured at θ= 0° corresponds to n0.67-n1 = 0.17 which, along with the measured Θ = 5.7°, gives Q 1 = 0.58 using Eq. (3). To cast light on the frequency dependence of the deflection, P(f,θ) ∝ E(f,θ)E *(f,θ) is depicted in Fig. 6(d) for THz pulses transmitted through the plasmonic router. Here, the normalized spectrums for pulses having appreciable signal level are shown. It is evident that the frequency content is similar for the THz pulses measured at various θ and, as such, the deflection is weakly dependent on f over the bandwidth of the transmitted pulses.
While the results introduced here effectively demonstrate the concept, it is expected that the deflection angle of the router could potentially be increased. As such, we will estimate the maximum attainable Θ. By examining Eq. (3), it can be seen that the parameters available are w, d, and Q 1. Clearly, decreasing w will increase Θ; however, w must remain large enough to accommodate several microparticles. Moreover, w must be large enough to accommodate the width of the transverse THz electric field profile, which is constrained by the diffraction limit. With these points in mind, a minimum value of w = 1 mm is postulated. Increasing d will also increase Θ, with a trade-off being increased absorption. While the amount of permissible absorption is subjective to the application and sensitivity of the apparatus, under the present experimental conditions absorption losses become unacceptable for d greater than 3.5 mm. Finally, it may be possible to increase Q 1 using a more sophisticated approach to form a varying τ C. However, the previously determined value Q 1 = 0.58 will be used since it is not practical to estimate the maximum possible value of Q 1. Therefore, with these considerations the maximum deflection is estimated to be ΘMAX = 20°. This deflection angle could be further amplified by placing a passive THz lens after the router. By using the same reasoning it is possible to estimate the minimum focal length for a plasmonic lens using Eq. (7). The input THz pulse should be collimated for focusing, which requires a transverse profile several wavelengths wide. As such, we assume a minimum width for a lens w = 3 mm. With d = 3.5 mm and assuming Q 2 = 0.58, we estimate a minimum focal length of F MIN = 3.2 mm. Note that this discussion only deals with Co and sapphire particles. If other materials are used, such as nickel and Teflon, the maximum deflection and minimum focal length would have to be reassessed.
To thoroughly examine the behaviour of the experimentally realized router, the polarization content of the transmitted THz radiation is considered. The various polarization states are experimentally accessed by rotation of the ZnSe crystal to an angle, Φ, relative to the THz electric field polarization. Angles Φ = 0°, 60° and 120° correspond to measurements of the horizontal polarization whereas Φ = 30° and 90° correspond to measurements of the vertical polarization . Shown in Fig. 7(a) is the experimentally obtained P for two perpendicular polarization states of the radiation transmitted through the device, where the horizontal (i.e. P H) and vertical (i.e. P V) polarization states are parallel and perpendicular to the linear polarization of the incident THz pulse, respectively. Here, it is apparent that the easured signals at the shown θ predominantly retain the linear polarization of the incident THz radiation. Nonetheless, for the horizontal and vertical polarization states the deflection angles are ΘH = 5.7° and ΘV = 6.9°, respectively. Clearly ΘV > ΘH, which can be understood by considering the fact that τ C decreases across the wavefront, where portions of the wavefront that pass through low-τ C regions interact with a greater number of dielectric particles and, consequently, polarization randomizing scattering events become more frequent. As depicted in Fig. 7(b), the portions of the wavefront that pass through regions with low τ C map to θ-values farther from 0° and, thus, the P V profile is centered farther from 0° compared to the P H profile.
The active behaviour of the device is investigated by applying B parallel to the polarization of the incident THz pulses. Here, P is measured for θ ranging from -10° to 20° and B values of 0 mT, 27 mT, 45 mT, 55 mT, 69 mT, and 78 mT, as shown in Fig. 8. It is clear from these results that for increasing B, the deflection of the THz pulse shifts towards θ = 0°. To elucidate this behaviour, Θ(B) is displayed in the inset of Fig. 8. As such, it can be seen that Θ decreases from 5.7° to 0.7° as B is increased from 0 mT to 78 mT. Therefore, the deflection can be accurately tuned to any angle within a 5° interval, or almost eliminated with an 88% efficiency for B = 78 mT. Moreover, ultrafast active operation using magnetic pulses may be possible, which would be important for practical considerations.
To better understand the magnetic response of the device, the frequency dependence of K 2(f, B) is calculated from the experimental data. Here, the frequency dependent Θ(f, B) is first obtained from the frequency dependent energy flux, P(f,θ,B), where
Here, it is noted that the FWHM of the spectrums span from 0.187 THz to 0.330 THz for B = 0 mT which decreases in width to span 0.149 THz to 0.240 THz for B = 78 mT. Such band limiting and red shift as B is increased are characteristic of photonic AMR . Moreover, the bandwidth for B = 0 is narrow compared to that of the incident THz pulses due to the band-limiting effects characteristic to plasmonic-enhanced transmission in random metallic media . Nevertheless, Eq. (6) is used to determine K 2(f, B), which is shown in Fig. 9(a) for B ranging from 0 mT to 78 mT and f ranging from 0.184 THz to 0.245 THz. To define a meaningful spectral range for all signals at various B, we choose the spectral region defined by the overlap between the spectral characteristic FWHM’s of the measured signals at the minimum and maximum B strengths (i.e. B = 0 mT and B = 78 mT). The weak dependence of K 2 on f permits the direct extraction of K 2(B) from the Θ(B) values in the inset of Fig. 8, as shown in Fig. 9(b). The fact that the maximum value of K 2(B) is 0.15, which is 88% of (n0.67-n1) = 0.17, sets the upper limit on the magnetically induced deflection of 88%.
Interestingly, we can use the experimentally determined values of K 2 to determine the expected magnetic dependence of the focal length (F) for a cylindrical plasmonic lens described by Eq. (11). Since the main constraint in the fabrication of the plasmonic devices is the imperfections in effective index profile offered through Q 1 and Q 2, it is reasonable to assume, without the loss of generality, that Q 1 ≈ Q 2 if the fabrication process is the same for the lens and router. This assumption is most accurate if the number and size of the constant-τ C regions used to create a variation of τ C from 0.67 to 1 is the same as for the router, which necessitates using w = 6 mm and d = 3.1 mm. As such, by using the K 2(B) regression line shown in Fig. 9(b), the expected F(B) is depicted in Fig. 9(c). Clearly, F undergoes a marked increase from 15 mm to 117 mm as B is increased from 0 mT to 80 mT. Since it has been experimentally shown that K 2(B) is weakly dependent on f, there would be negligible chromatic aberrations introduced by the application of B.
As a final note, we address the attenuation of the THz pulses transmitted through this type of plasmonic device. It can be seen from Fig. 6(a) that passage of the THz pulse through the plasmonic router results in a factor of 40 reduction in transmitted energy flux. A similar figure is expected for a plasmonic lens. Approximately an order of magnitude of the loss results from reflection at the input plane of the device due to the high concentration of metallic particles, while the remaining losses stem from absorption in the device . While this loss may be a limiting factor for some applications, for active operation the primary concern would be any losses that result from the application of B. For example, unlike a THz passive lens used to enhance intensity, for an active THz lens the absolute intensity at the focal spot may not be as important compared to the ability to actively tune the size of the focal spot and the focal length. For B = 78 mT, the magnetic dependent loss is less than 20% relative to the energy flux when B = 0.
This work introduces a class of active plasmonic devices that can be tailored to fulfill various roles, such as active plasmonic directional routers or active plasmonic lenses. Here, the transmission through the devices is mediated by near-field coupling of particle plasmon modes between neighbouring ferromagnetic particles. Furthermore, we show that a position dependent phase accumulation can be actively modulated via magnetic field dependent plasmonic coupling due to spin-orbit interaction in ferromagnetic media. This type of plasmonic device is explored through both numerical calculations and an experimental study. In the former case, the passive operation of directional router and cylindrical lens geometries are investigated, whereas in the latter case the active, B-dependent operation of a directional router is observed. The active control over the propagation of THz radiation that is enabled by this work may be applicable in active photonic systems.
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and Canada Research Chairs (CRC).
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