ar X iv :1 50 7. 08 16 7v 2 [ ph ys ic s. op tic s] 1 2 O ct 2 01 5 Phase transition in PT symmetric active plasmonic systems M. Mattheakis1,2, T. Oikonomou1,3, M. I. Molina4, G. P. Tsironis1,3,5 1Crete Center for Quantum Complexity and Nanotechnology, Department of Physics, University of Crete, PO Box 2208, 71003 Heraklion, Greece 2School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA 3Department of Physics, Nazarbayev University, 53 Kabanbay Batyr Ave., Astana 010000, Kazakhstan 4Departamento de F́ısica, Facultad de Ciencias, Universidad de Chile, Santiago, Chile 5National University of Science and Technology MISiS, Leninsky prosp. 4, Moscow, 119049, Russia (Dated: July 30, 2018) Surface plasmon polaritons (SPPs) are coherent electromagnetic surface waves trapped on an insulator-conductor interface. The SPPs decay exponentially along the propagation due to con- ductor losses, restricting the SPPs propagation length to few microns. Gain materials can be used to counterbalance the aforementioned losses. We provide an exact expression for the gain, in terms of the optical properties of the interface, for which the losses are eliminated. In addition, we show that systems characterized by lossless SPP propagation are related to PT symmetric systems. Fur- thermore, we derive an analytical critical value of the gain describing a phase transition between lossless and prohibited SPPs propagation. The regime of the aforementioned propagation can be directed by the optical properties of the system under scrutiny. Finally, we perform COMSOL simulations verifying the theoretical findings. Keywords: Surface plasmons polaritons, active/gain materials, PT symmetry, lossless propagation. I. INTRODUCTION A light-matter interaction called surface plasmon po- laritons (SPPs) has gained the scientists’ interest due to its unique properties, such as control of electromag- netic energy in subwavelength scales [1–4], high sensi- tivity in dielectric properties [3, 5, 6], negative refraction and hyperbolic wave front [7, 8]. SPPs have been applied in nanophotonics, imaging, optical holography, nano an- tennas, biosensing, integrated circuits and metamaterials [6, 9–12]. Important progress has been made in plasmon- ics with two-dimensional materials, such as graphene and black phosphorus, where the plasmonic properties can be tuned by using chemical doping or applying external gate voltage [3, 13–18]. Moreover, plasmonic lenses, waveg- uides and meta-materials based on graphene have already been applied [3, 19, 20]. Last but not least, multi-layers structures have been created by stacking two-dimensional crystals one on top of another providing surprising elec- tronic and optical features [14, 18, 21]. Near plasma frequency ωp, the electrons on the sur- face of metals or semiconductors are free to move sus- taining collective oscillations [2, 22–26]. The coupling between light and electron oscillations allows the creation of Transverse Magnetic (TM) Electromagnetic (EM) sur- face waves, namely surface plasmon polaritons (SSPs). From the mathematical point of view, SPPs are surface waves bounded along the interface between two mate- rials with sign reversed dielectric permittivities, i.e. a dielectric-conductor interface, and their EM field decays exponentially away from the interface (evanescent waves) [2, 22–26]. In this work, we focus on plasmonic waveguides formed by a planar interface which consists of two semi-infinite layers with reversed sign permittivities, namely a dielec- tric and a metal. The dispersion relation which char- acterizes the SPPs propagation can be determined by Maxwell Equations (ME) as [2, 4, 23, 24] β = k0 n = k0 √ εdεm εd + εm , (1) where k0 = ω/c is the free space wave number of the incident excitation light of angular frequency ω, n is the plasmon effective refractive index, εd and εm the permit- tivity of dielectric and metal, respectively, and c is the speed of light in vacuum. SPPs decay exponentially along the interface as well, due to the metal losses. In mathematical language, the metal losses are described by a negative imaginary part in the permittivity function of the metal, i.e. εm = −ε1m − iε2m, where ε1m, ε2m > 0. Consequently, the SPPs wave number β becomes complex, viz. β = β′+iβ′′, where the imaginary part accounts for losses of SPPs en- ergy. The imaginary part ℑ[β] of Eq. (1) determines the characteristic propagation length L, which shows the rate of change of the energy attenuation of SPPs along the propagation axis [4, 23, 24], that is L = 1 2ℑ[β] . (2) Gain materials, rather than passive dielectrics, have been used to reduce the losses in SPPs propagation. These active materials are characterized by a complex permittivity function, i.e. εd = ε1d+iε2d with ε1d, ε2d > 0, where the imaginary part accounts for gain, that is, the dielectrics give energy to the system counterbalanc- ing the metal losses [4, 23, 27–29]. In addition, active dielectrics have been used for exploring PT symmetry in optical systems [30–33] characterized by the condition that n(−x) = n∗(x), where n and n∗ the refractive in- dex and its complex conjugate, respectively; x denotes http://arxiv.org/abs/1507.08167v2 2 the spatial coordinate along the interface. Metamateri- als with PT symmetric effective refractive index can be constructed by the combination of gain dielectrics and loss metals [31–33]. What makes PT symmetric media interesting is that they allow control over EM field by tuning the gain and loss of the materials. It has been already demostrated in [4, 23, 28, 29] that for a certain value of gain, the losses in SPPs propagation may vanish. Consequently, the SPPs propagation con- stant β as well as the effective refractive index n become real and therefore the PT symmetry is satisfied, since n does not exhibit any spatial dependence along the inter- face. Furthermore, Eq. (1) states that a PT symmetric n leads to infinite propagation length, viz. lossless SPPs propagation. In the present work, we investigate theoretically and numerically the PT symmetry in active plasmonic sys- tems. In Section II, we provide an explicit expression of the gain, namely εPT , for which the losses in SPPs propagation have been eliminated. In addition, we find a critical value εc of εPT , where SPPs wave number β and the effective SPPs refractive index n shift from real to imaginary regime, subsequently the εc is a PT sym- metry breaking point. It is remarkable that it is a steep phase transition from lossless to prohibited SPP propa- gation, which offers the opportunity to control whether SPPs propagate or not by tuning the optical properties of the interface. In Section III, we apply the theoreti- cal results derived in the previous Section on interfaces comprised of active dielectrics and Drude metals. In Sec- tion IV, we proceed with numerical simulations by solv- ing the full system of ME in the frequency domain by using the commercial multiphysics software COMSOL, and we show that lossless SPPs propagation correspond- ing to PT symmetry can be achieved in the presence of gain dielectrics. Finally, concluding remarks are offered in Section V. II. PT AND CRITICAL GAIN In this Section, we calculate the exact expression of the dielectric permittivity gain counterpart ε2d, for which the SPPs propagate without losses in the dielectric-metal interface. Plugging the complex structure of the dielectric and metal permittivity into Eq. (1), function n can be written in the ordinary complex form as [34] n = √ √ x2 + y2 + x 2 + i sgn(y) √ √ x2 + y2 − x 2 (3) where sgn(y) is the discontinuous signum function and x := ε1d‖εm‖2 − ε1m‖εd‖ 2 ‖εd + εm‖2 (4a) y := ε2d‖εm‖2 − ε2m‖εd‖ 2 ‖εd + εm‖2 (4b) with ‖ε∗‖ denoting the norm of the complex number ε∗. Considering the plasmon effective index n in Eq. (3) in the (x, y)–plane, we observe that a lossless SPP prop- agation, i.e, ℑ[β] = 0, is warranted when the conditions y = 0 and x > 0 are simultaneously satisfied. For y = 0 and x < 0, although the imaginary part in Eq. (3) van- ishes due to the signum function, its real part becomes imaginary, i.e. β = i √ |x|, which does not correspond to propagation SPP modes. Studying the permittivity dependence of x and y in Eq. (4) and solving the condi- tion y = 0 with respect to the dielectric gain part ε2d for εd 6= −εm, we obtain two exact solutions, i.e., ε2d → ε± 2d of the form ε± 2d = ‖εm‖2 2ε2m  1± √ 1− ( 2ε1dε2m ‖εm‖2 )2   (5) The result in Eq. (5) is in agreement with the one de- rived in Refs. [4, 23] following yet a different derivation path. Invoking the physical argument of the SPP wave bound to the dielectric-metal interface, we read that only ε− 2d is of physical relevance, since ε+ 2d leads to waves ra- diating in the transverse towards the interface direction [23]. Taking into account the, by definition, positive real domain of ε− 2d and the dependence of the latter on the metal-dielectric components, we read that the following inequality has to be satisfied ‖εm‖2 > 2 ε1d ε2m (6) For εd = −εm both x and y diverge exhibiting asymp- totically the same image, thus lim εd→−εm ℜ[β] = lim εd→−εm ℑ[β] → +∞ (7) This in turn means that the former complex point does not belong to the set domain of the lossless SPP propa- gation since y 6= 0. Solving, on the other hand, the equation x = 0 with respect to the dielectric gain ε2d for εd 6= −εm, we may determine the critical value εc distinguishing the regimes of lossless and prohibited SPP propagation, namely εc = ε1d √ ‖εm‖2 ε1mε1d − 1 (8) Equating Eqs. (5) and (8), i.e., y = 0 = x, we obtain the condition ε1d ε2m (ε1d − ε1m) = 0 which reduces to ε1d = ε1m, since ε1d, ε2m > 0. Replacing the former value of ε1d in Eqs. (5) and (8) we obtain ε− 2d 6= εc and ε− 2d = εc = ε2m for ε1m < ε2m and ε1m > ε2m, respec- tively. The former case is obviously a contradiction. The latter case corresponds to the singularity point in Eq. (7), where x, y 6= 0, thus it is a contradiction as well. In other words, x and y do not become zero simultaneously, implying that the critical value εc is not an element of the domain set of the wave number β(y = 0) ≡ β0. This is in agreement with the propagation length L in Eq. 3 (2). Indeed, when y = 0 then L tends to infinity, which means that the SPP wave number must exhibit a nonzero value. An even more interesting point, unveiled from the y = 0 = x analysis, is the estimation of the β0 behaviour when approaching the critical point, ε− 2d → εc (y = 0 and x → 0), described by Eq. (7) for β → β0 with ε1m > ε2m. Physically, the former point corresponds to the wave elec- trostatic character of zero phase velocity, known in lit- erature as the surface plasmon mode [24]. Including the discontinuity at the point εc, the entire codomain of β0 is described as follows β0 =      Real, x > 0 Imaginary, x < 0 Complex Infinity, x → 0 (9) In the general case of β, where y (excluding the point εd = −εm) may take nonzero values as well, we observe the following. For ε2d < ε− 2d, the signum function in Eq. (3) is negative since then y < 0 implying ℑ[β] < 0 ⇒ L < 0. This means that the imaginary part of β accounts for losses and the SPP amplitude decreases along the propagation surface. Reversely, for ε− 2d < ε2d < ε+ 2d yielding y > 0 ⇒ sgn(y) > 0, we have ℑ[β] > 0 ⇒ L > 0. In this case the imaginary part of β accounts for gain and the SPPs amplitude increases along the propagation surface. In the special case of y = 0 ⇒ ε2d = ε− 2d studied above, the SPP amplitude is constant along the propagation surface. This behaviour of the signum function fully explains the results observed in Ref. [23] regarding the SPP amplitude. An interesting feature of the lossless SPP propagation case, i.e., for ε−2m < εc, in regard to the refractive index n is that the latter fulfils the condition n(y = 0) = n∗(y = 0), since its imaginary part vanishes owing to the signum function. This in turn, may be considered as the PT symmetry phase condition, where n is spatial indepen- dent. However the structure is not PT symmetric in the narrow sense, the real value of the supported propagation constant along the interface admits time-reversal and ge- ometrical symmetry. Then, the dielectric gain expression ε− 2d in Eq. (5) can be attributed to the PT symmetry property satisfied by the lossless SPP propagation and denoted as ε− 2d ≡ εPT . We shall keep this denomination in what follows. On the contrary, in the case ε−2m < εc, the PT condition is not satisfied, since the refractive in- dex is imaginary. Subsequently, the critical gain εc may be regarded as the PT -symmetry breaking point of the plasmonic system under scrutiny. III. ACTIVE DIELECTRIC - DRUDE METAL INTERFACE It is quite remarkable that εPT as well as εc depend on the optical properties of the dielectric and metal, that is, ε1d, ε1m and ε2m. The metal permittivity in turn may generally exhibit a dependence on the angular frequency ω, so that by tuning ω we may control the values of εPT lying below or above εc. Precisely, for interfaces comprised of an active dielectric and a Drude metal [24], εm is given as εm(ω) = − ( ω2 p ω2 + Γ2 − εh ) − i ω2 pΓ ω(ω2 + Γ2) (10) where εh denotes the high frequency permittivity, ωp is the plasma frequency and Γ accounts for metal losses in frequency units [24]. By virtue of Eq. (10) we can express the εPT and εc in terms of the frequency ω. Moreover, taking Eq. (7) into consideration, we may obtain the SPP resonance frequency ωsp [2, 4, 24] ωsp = √ ω2 p ε1d + εh − Γ2 (11) It can be proven that for Drude metals the εPT is always smaller than εc for frequencies lower than ωsp. Thus, according to our theoretical results we anticipate loss- less SPPs propagation for ω < ωsp and prohibited SPPs propagation for ω > ωsp. In order to verify our theoretical predictions, we cal- culate the SPP dispersion relation for an interface con- sisting of silver with εh = 1, ωp = 1.367 1016Hz, Γ = 1.018 1014Hz and silica glass with ε1d = 1.69 and ε2d = εPT . The frequency values are confined in the regime imposed by the inequality in Eq. (6). In Fig. 1 we plot the real (dotted blue line) and imaginary (or- ange dashed line) part of the normalized SPP dispersion relation β0/kp (kp ≡ ωp/c) with respect to the normal- ized frequency ω/ωp; the yellow dash-dot line shows the wave number (k-number) in the dielectric, and the res- onance frequency ωsp is represented by the horizontal green solid line where the interchange between ℜ[β] and ℑ[β] appears. We observe, indeed, that for ω < ωsp the imaginary part of β vanishes while for ω > ωsp the SPPs wave number is purely real. Subsequently, in the vicinity of ω = ωsp a phase transition from lossless to prohibited SPPs propagation is expected (see Section IV). Fig. 1 highlights the relation between β and the metal permittivity, and demonstrates the PT symme- try breaking point εc, where the ℜ[β] vanishes. In Fig. 2 we consider a variable ε1d and record the depen- dence on it of both the magnitude ℜ[β] in Eq. (1) and εPT in Eq. (5) for three different frequencies, namely ω = {0.45ωp, 0.5ωp, 0.55ωp}. The former is represented by color lines on the complex plane defined by (ε1d, εPT ) for x and y axis, respectively. Each color line corresponds to a different frequency. Fig. 2 unveils that the more dense the dielectric is the higher value of the gain we need for having undamped SPPs propagation. In ad- dition, the ℜ[β] vanishes very suddenly as we increase the gain, verifying that at this point the PT symmetry breaks and the SPPs propagation becomes prohibited. According to the aforementioned figure we can tune the 4 β /k p 0 2 4 6 8 ω /ω p 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 ℜ[β] ℑ[β] k-number in dielectric ω sp FIG. 1: The SPP dispersion relation β under PT symmetry (ε2d = εPT ), i.e., β0, with respect to the frequency ω. ℜ[β] and ℑ[β] are indicated by dotted blue line and orange dashed line respectively. The dash-dot yellow line is referred to the wave number of light in the dielectric, whereas the horizontal solid green line shows the SPP resonance frequency ωsp where the interchanging between ℜ[β] and ℑ[β] appears. kp = ωp/c is used as normalized unit of wavenumbers and ωp as normal- ized unit for frequencies as well. magnitude of the ℜ[β] as well as εc and εPT by choosing the appropriate dielectric. IV. SIMULATIONS In this Section, we verify our theoretical predictions of Sections II and III, by solving numerically the full sys- tem of ME in the frequency domain in a two dimensional space (2D) for TM polarization electric and magnetic fields. The numerical experiments have been performed by virtue of the multi-physics commercial software COM- SOL. Precisely, we explore the SPPs propagation length L with respect to ω on the interface between two semi- infinity layers, i.e., an active dielectric and a Drude metal, recording the desired phase transition from lossless to prohibited SPP propagation. We further demonstrate the lossless SPPs propagation, analysing the magnetic field intensity along the surface of two known in literature configurations, the Kretschmann-Raether and the Otto configurations [2, 24, 29]. In our numerical experiments the frequency ω is confined in the range [0.3ωp, 0.75ωp] with the integration step ∆ω = 0.01ωp. Regarding the active dielectric – Drude metal interface described in the previous section, we conduct the near- field excitation technique [15, 16, 24] to excite SPPs on the metallic surface. For this purpose, a circular EM source of radius R = 20nm has been located 100nm above the metallic surface acting as a point source, since the wavelength λ of the EM wave in the silica glass is con- 1d 1 2 3 4 5 P T 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 ℜ[β]/k p 1 2 3 4 5 6 7 ω=0.45 ω p ω=0.5 ω p ω=0.55 ω p FIG. 2: The PT gain εPT with respect to the real part of the dielectric ε1d is plotted for three different frequencies ω. The colors on each curve represents the magnitude of the ℜ[β] nor- malized to kp. We observe that the more dense the dielectric is the more gain is needed to achieve lossless SPP propaga- tion. For all frequencies there is a critical gain for which the ℜ[β] transmits suddently from high magnitude (bright col- ored line) to zero value (black line), showing gain saturation and a phase a transition from undapted SPPs to forbidden propagation. strained to λ >> R [24]. In addition, Perfectly Matched Layers (PML) are used as boundary conditions. In Fig.3 we demonstrate, in a log-linear scale, the prop- agation length L with respect to ω subject PT symmetry (blue line and open circle). For the sake of comparison, we plot L(ω) for the gainless case (green line and filled circles). The solid lines represent the theoretical predic- tions obtained by Eq. (2), whereas the circles indicate COMSOL results. For the numerical calculations, the characteristic propagation length has been estimated by the inverse of the slope of the log(I), where I is the mag- netic intensity along the interface [2, 24, 35]. The red vertical dashed line denotes the SPP resonance frequency ωsp, in which the phase transition appears. The graphs in Fig. 3 indicate that in the presence of the PT gain, i.e. ε2m = εPT , the SPPs may travel for very long, practi- cally infinite, distances. Approaching the resonance fre- quency ωsp, L decreases rapidly leading to a steep phase transition on the SPPs propagation. The deviations be- tween theoretical and numerical results in Fig. 3 for fre- quencies near or greater than ωsp are attributed to the fact that in the regime ωsp < ω < ωp, there are quasi- bound EM modes [24], where EM waves are evanescent along the metal-dielectric interface and radiate perpen- dicular to this. Consequently, the observed EM field for ω > ωsp does not correspond to SPPs but belongs to the quasi-bound modes. So far, the theoretical findings in Sections II and III have been successfully confirmed. We further pro- 5 ω/ω p 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 P ro pa ga tio n Le ng ht L ( m ic ro ns ) 10 -2 10 0 10 2 10 4 L PT (theory) L PT (comsol) L No gain (theory) L No gain (comsol) ω sp FIG. 3: The SPP characteristic length L is demonstrated as function of the frequency ω subject PT symmetry (blue line/open circles) and in the case of gainless propagation (green line/points). In both cases, the solid lines indicate the theoretical prediction, whereas the points show the nu- merical results obtained by COMSOL simulations. A phase transition from lossless (large L) to prohibited propagation (small L) occurs in the case of PT symmetry. The red verti- cal dashed line indicates the resonance frequency ωsp, where the phase transition takes place. The theoretical curves show deviation from simulations for frequencies near and greater that ωsp, because in this regime quasi-bound EM modes ap- pear. ceed investigating the PT symmetry in active plasmonic systems. We perform COMSOL simulations based on the Total Internal Reflection (TIR) method, applied on the Kretschmann-Raether and Otto configurations, sepa- rately. Within the former, a thin metal film is sandwiched between two dielectrics with the incident wave hitting the denser medium. In Otto configuration, the denser the di- electric and the metal sandwich a lighter dielectric. In both configurations a type of silica glass, with dielectric constant εD = 4, is used as denser passive dielectric, whereas for active dielectric (lighter medium) as well as for metal, we use the materials described in Section III. Furthermore, for the COMSOL simulations [35, 36], we utilize a monochromatic plane wave source of frequency f = 870 THz, amounting to 40% of plasma frequency ωp, and corresponding to a wavelength λ = 345nm in the ultraviolet regime. Again, PML are used as bound- ary conditions. According to the Eq. (5), the PT gain for which SPPs propagate without losses is calculated to be εPT = 0.012. We observe, however, that in the numerical experiments a larger gain is needed (of the same magnitude though), namely ε̃PT = 0.026, for both configurations. This devia- tion from the theoretical value can be justified if one takes into consideration that the assumption of semi-infinitely thick metal and dielectric layers composing the interface [23], under which the SPP dispersion relation of Eq. (1) holds true, is experimentally not fully satisfied. (a) (b) FIG. 4: COMSOL simulations show the intensity distribution of the magnetic field on a Kretschmann-Raether configura- tion, for (a) ε2d = 0 (No gain) (b) ε2d = ε̃PT = 0.026 (PT symmetry). In the Kretschmann-Raether configuration a thin metal of thickness d = 45nm has been used for exciting SPPs. The resulting propagation is illustrated in Fig. 4a under lack of gain and in Fig. 4b for the PT symmet- ric case. The corresponding profiles of the magnetic field intensity along the interface are demonstrated in Fig. 5, where the lossless SPP propagation is evident. x ( µ m) 0 5 10 15 20 25 30 In te n si ty ×10 4 1 2 3 4 5 No Gain x ( µ m) 0 5 10 15 20 25 30 In te n si ty ×10 5 1 2 3 PT gain (a) (b) FIG. 5: Characteristic profile of the magnetic field intenstity along the interface at Kretschmann-Raether configuration of Fig. 4, for (a) ε2d = 0 (No gain) (b) ε2d = ε̃PT = 0.026, where lossless SPP propagation is achieved. In the Otto configuration, on the other hand, the SPPs 6 excitation is succeeded by means of an active dielec- tric of thickness d = 150nm which has been used be- tween a non-active dielectric and a metal. By analogy to the Kretschmann-Raether configuration experiment, we present the SPP propagation without gain in Fig. 6(a) and for the PT gain with ε̃PT = 0.026 in Fig. 6(b). In Fig. 7 the corresponding profiles of the magnetic inten- sity along the interface are presented, unveiling again a clear lossless SPPs propagation in the PT case. (a) (b) FIG. 6: COMSOL results for the intensity distribution of the magnetic field at an Otto configuration, for (a) ε2d = 0 (No gain) (b) ε2d = ε̃PT = 0.026 (PT symmetry). x ( µ m) 5 10 15 20 25 30 In te n si ty 5000 10000 15000 No Gain x ( µ m) 5 10 15 20 25 30 In te n si ty 2000 4000 6000 8000 10000 12000 PT gain (a) (b) FIG. 7: Characteristic profile of the magnetic field intenstity along the interface at Otto configuration of Fig. 6, for (a) ε2d = 0 (No gain) (b) ε2d = ε̃PT = 0.026, where lossless SPP propagation is achieved. V. CONCLUDING REMARKS Summarizing, we have investigated the role of ac- tive/gain dielectrics in plasmonic systems. In particular, we have studied the propagation properties of surface plasmon polaritons (SPPs) along an interface confined by two semi-infinite layers: a dielectric and a metal. We have calculated an exact expression ε− 2d for the dielectric gain ε2d, for which the metal losses have been completely counterbalanced, resulting to lossless SPPs propagation along the interface. We argued that the a plasmonic sys- tem characterized by the aforementioned lossless prop- agation may be related to PT symmetric systems, i.e., εPT ≡ ε− 2d. Within the PT symmetry, a critical gain εc exists distinguishing between the real and imaginary part of the SPP dispersion relation. This distinction cor- responds to a phase transition from lossless to prohibited SPP propagation. It is remarkable that the εPT as well as the εc depend on the optical properties of the interface. We applied our theory to interfaces consisting of Drude metals and gain dielectrics demonstrating the predicted by the theory lossless propagation as well as the phase transition at the SPP resonance frequency ωsp. We per- formed numerical simulations with COMSOL software, using the near-field excitation method in order to inves- tigate our theory, verifying successfully all the theoretical predictions. We also performed COMSOL simulations for two different plasmonic configurations based on the TIR method- Kretschmann-Raether and Otto configurations- where lossless SPP propagation can be achieved. Active metamaterials may be designed to have the desirable frequency-dependent permittivity response, as Eq. (5) points out; these metamaterials could be used for the fabrication of PT symmetric plasmonic systems, providing infinite SPPs propagation. The active meta- materials may be used to design PT symmetric plas- monic integrated circuits which could transfer informa- tion in sub-wavelength scales for large (theoretically in- finite) distance, rather than a passive plasmonic system where SPPs propagate for few micrometers. Moreover, we demonstrated that there is a threshold in the PT gain values, above which the PT symmetry breaks and thus the system passes from lossless to prohibited propaga- tion. The gain threshold as well as the PT gain depend on the optical properties of the dielectric and metal, sub- sequently we could control the SPPs propagation by tun- ing the dielectric constant of metal εm or the real part of dielectric permittivity ε1d; for instance, the former, i.e. εm, is usually frequency-dependent, thus we can inter- change between lossless and prohibited SPP propagation by tuning the frequency of the incident EM wave. Acknowledgements This work was supported in part by the European Union program FP7-REGPOT-2012-2013-1 under grant agreement 316165. 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