Pulsed EM radiation from a traveling-current plasmonic nanowire

Photonics and Nanostructures – Fundamentals and Applications 22 (2016) 35–39 Contents lists available at ScienceDirect Photonics and Nanostructures – Fundamentals and Applications jou rn al hom epage: www.elsevier.com/locate/photonics Pulsed EM radiation from a traveling-current plasmonic nanowire Martin ˇ Stumpfa,∗, Guy A.E. Vandenboschb aSIX Research Centre, Brno University of Technology, Technická 3082/12, 616 00 Brno, Czech Republic bESAT – TELEMIC Division, KU Leuven, Kasteelpark Arenberg 10 bus 2444, 3001 Leuven, Belgium a r t i c l e i n f o Article history: Received 3 June 2016 Received in revised form 15 September 2016 Accepted 6 October 2016 Available online 21 October 2016 Keywords: Plasmonic Nanowire segment Time domain EM radiation a b s t r a c t Pulsed electromagnetic (EM) radiation from a traveling-current plasmonic-wire segment is studied analytically using the unilateral Laplace-transform technique. This approach yields closed-form expressions that can be readily evaluated for given configurational and excitation parameters, thereby revealing physical insight into the time-domain (TD) EM radiation behavior of a plasmonic nanowire. Illustrative numerical examples concerning pulsed EM fields radiated from a gold nanowire are given and discussed. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction Antennas capable of efficiently operating with optical wave fields show a lot of promise because of their applications in THz and photonic devices [1,2] and plasmonic biosensors [3], for example. Notwithstanding the applicability of some general-purpose numerical EM solvers (see e.g. [4]), the still increasing complexity of plasmonic structures has necessitated the development of dedicated (and more efficient) numerical methodologies [5–7]. As to the corresponding TD (i.e. space–time) modeling, this category is (almost) exclusively limited to the finite-difference time-domain (FDTD) technique (see [8] and [9, Ch. 4], for example). Although the FDTD technique is a well-established tool for engineering practice, its purely numerical outcomes can hardly be sufficient to fully grasp all peculiarities of plasmonic phenomena. The latter can be best addressed by solving canonical problems such as the excitation of surface plasmon polaritons at planar interfaces [10, Ch. 2]. Despite the fact that all physical phenomena manifest themselves in space–time, only a few initial attempts to describe plasmonic effects analytically in TD do exist so far (see [11–14], for example). Except for the observation that the skin depth cannot be neglected anymore [15], analytical models characterizing frequency-domain (FD) EM scattering from optical plasmonic nanowires (see [16], for example) rely largely on the classic FD ∗Corresponding author. E-mail addresses: [email protected], [email protected] (M. ˇ Stumpf). theory of straight-wire antennas [17]. To the best of our knowledge, there is presently no study available that analyzes the pulsed EM radiation from a plasmonic nanowire analytically. Filling this void is hence the main purpose of this work. This paper follows in part the methodology based on the unilateral Laplace transformation that has been successfully applied to analyzing a relaxation-free traveling-current straight-wire segment in TD [18]. Here it is demonstrated that such a methodology is also applicable to describing the pulsed EM radiation from a current pulse traveling along a plasmonic nanowire. Indeed, it is shown that the pulsed EM radiation characteristics of such a radiating segment can be expressed as the superposition of the EM radiation characteristics pertaining to the corresponding electrically perfectly-conducting (PEC) wire and the (Boltzmann-type) relaxation part describing its plasmonic behavior. For an earlier work on pulsed EM radiation from a travelingwave PEC antenna we refer the reader to [19]. Finally, a somewhat more general TD approach accounting for the complete space–time electric-current distribution along a thin PEC wire can be found in [20]. 2. Problem definition The plasmonic nanowire under consideration is shown in Fig. 1. The wire is placed in the unbounded, homogeneous and isotropic embedding of permittivity 0and permeability 0. The corresponding EM wave speed is c0= (00)−1/2 > 0 and 0= (0/0)1/2 > 0 is the free-space impedance. The EM properties of the wire itself are described by the radial plasma frequency http://dx.doi.org/10.1016/j.photonics.2016.10.002 1569-4410/© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4. 0/). 36 M. ˇ Stumpf, G.A.E. Vandenbosch / Photonics and Nanostructures – Fundamentals and Applications 22 (2016) 35–39 Fig. 1. Problem configuration. ωpand the collision frequency cvia the conduction relaxation function (see [21, Section 19.5] and [11]) (t) = 0ω2 pexp(−ct)H(t)(1) in which H(t) is the Heaviside unit-step function. Finally, the wire length is denoted by . The wire segment is at z = 0 and t = 0 excited by an electriccurrent pulse I(t) that travels along the segment’s axis. We assume that I(t) = 0 for t < 0 along with the zero initial conditions for all EM fields throughout the problem configuration. Similarly to [19], we shall primarily limit ourselves to describing pulsed EM radiation of the electric-current pulse as it traverses the plasmonic segment from z = 0 to z =. Beyond this limitation, reflections at the antenna end points have to be properly accounted for. This fact is demonstrated on a numerical example given in Section 5. The spatial point where the pulsed EM radiation is observed is specified by the radial, azimuthal and axial coordinates {r, , z}, respectively, with respect to the Cartesian reference frame with the origin O. The time coordinate is t. Partial differentiations are denoted by ∂ with the corresponding subscript. The time-integration and timeconvolution operators are denoted by ∂−1 tand *, respectively. 3. Radiated-field source-type representations Owing to the rotational symmetry of the plasmonic-wire configuration, the non-zero EM-field components are -independent and can be expressed through (cf. [21, Section 26.4]) Er(r, z, t) = −1 0∂−1 t∂r∂zAz(r, z, t) (2) Ez(r, z, t) = −0∂tAz(r, z, t) + −1 0∂−1 t∂2 zAz(r, z, t) (3) H(r, z, t) = −∂rAz(r, z, t) (4) in which Azcan be, symbolically, cast into the following form (cf. [18, Eq. (3)]) Az(r, z, t) = I(t) ∗ (r, z, t) (5) where we have assumed the thin-wire approximation and (the complex-frequency-domain counterpart of) (r, z, t) will be specified below. Under a unilateral Laplace transformation ˆ Az(r, z, s) =∞ t=0 exp(−st)Az(r, z, t)dt (6) with the complex-frequency parameter {s ∈ C; Re(s) > 0}, Eq. (5) can be transformed as follows: ˆ Az(r, z, s) =ˆ I(s)ˆ (r, z, s) =ˆ I(s) =0 exp −sR()/c0/4R() exp −s2+s/(s + c)ω2 p1/2/c0d (7) with R() = [r2+ (z − )2]1/2. In Eq. (7) we may distinguish between the propagation factors exp(−sR/c0) and exp(− ˆ) with  ∈ (0, ) pertaining to the wave propagation in the embedding and along the wire segment itself, respectively. In accordance with Eq. (1), the propagation coefficient corresponding to the plasmonic wire can be found as ˆ = {s2+ [s/(s + c)]ω2 p}1/2/c0(see [21, Eqs. (24.4-13), (24.4-14) and (26.2-3)]). Upon expanding R() about R(0) = (r2+ z2)1/2→ ∞ we arrive at the following far-field approximation ˆ Az(r, z, s) =ˆ A∞ z(, s) exp −sR(0)/c0/4R(0) 1 + O[R−1(0)](8) in which ˆ A∞ z(, s) =ˆ I(s) =0 exp s cos()/c0 exp −[s2+ [s/(s + c)]ω2 p]1/2/c0d (9) Along these lines, the transient EM radiation characteristics can be then expressed using the TD counterpart of (9) with (2)–(4) as follows: E∞ r(, t) = 0∂tA∞ z(, t) sin() cos() (10) E∞ z(, t) = −0∂tA∞ z(, t)sin2() (11) H∞ (, t) = c−1 0∂tA∞ z(, t) sin() (12) with (cf. Eq. (8)) Er(r, z, t) = E∞ r, t − R(0)/c0/4R(0){1 + O[R−1(0)]} (13) R(0) → ∞, for example. Finally note that the -component of the electric-type radiation characteristic directly follows as E∞ (, t) = 0∂tA∞ z(, t) sin()(14) which gives E∞ /H∞ = 0for all t > 0 and {0 <  ≤ }. From Eqs. (10)–(12) and (14) it is clear that the transient radiation characteristics of the analyzed plasmonic wire are proportional to ∂tA∞ z(, t). Accordingly, the main subject of the following section is to find (the time-derivative of) the TD counterpart of Eq. (9). 4. Pulsed EM radiation characteristics Owing to the availability of low-loss plasmonic materials such as gold or silver (see e.g. [22]), we will, in the first approximation, limit ourselves to the collision-free case by taking the limit c↓ 0. In this case, the solution is attainable in terms of standard functions. After some straightforward steps we end up with A∞ z(, t) = A∞;PEC z(, t) −c0ωp [1 − cos()]2I(t) ∗min[t,T()] =0 J1ωp[t − ]1/2[t + q()]1/2d [t − ]1/2[t + q()]1/2(15) where T() = (/c0)[1 − cos()], q() = [1 + cos()]/[1 − cos()], J1(x) is the Bessel function of the first kind and of the first order and A∞;PEC z M. ˇ Stumpf, G.A.E. Vandenbosch / Photonics and Nanostructures – Fundamentals and Applications 22 (2016) 35–39 37 Fig. 2. Excitation electric-current pulse shape. corresponds to the instantaneous response of the corresponding PEC wire segment, viz A∞;PEC z(, t) =c0I(t) 1 − cos()∗H(t) − H[t − T()](16) with A∞;PEC z(0, t) = I(t)  H(t). Now, upon using (15) and (16) in Eq. (14) we may specify the -component of the electric-type radiation characteristics, viz E∞ (, t) = E∞;PEC (, t) −0ωpsin() [1 − cos()]2∂tI(t) ∗min[t,T()] =0 J1ωp[t − ]1/2[t + q()]1/2d [t − ]1/2[t + q()]1/2(17) where E∞;PEC (, t) = 0I(t) − I[t − T()]sin() 1 − cos()(18) is the field that corresponds to the PEC wire segment, with E∞;PEC (0, t) = 0. Similarly to the Hertzian dipole whose radiated amplitude is proportional to the time derivative of the excitation electric-current pulse [21, Section 26.9], also the far-field amplitude given by Eq. (18) is bipolar and is composed of the (scaled) electriccurrent pulse itself and its negative and delayed copy. The far-field radiated amplitude of the plasmonic wire segment described by Eq. (17) in addition includes the relaxation part whose contribution can be strongly oscillatory, depending on the parameter  ωp/c0and the relative excitation-pulse time width. The radial and axial components of the radiation characteristics can be found along the same lines. 5. Numerical results The results derived in the previous section are applied to analyze the pulsed EM radiation from a gold (Au) wire segment of length =500 (nm). For the following calculations the plasma frequency is ωp= 8.55 (eV) [22]. Note that these parameters give ωp/c0≃ 21.7, for which the oscillatory integral in Eq. (17) is still easily manageable via standard integration routines such as the trapezoidal rule, for example. The radiating nanowire is activated via the electriccurrent pulse with the power-exponential signature [18] I(t) = Imt/trexp −t/tr− 1H(t) (19) where we take Im= 1.0 (A), c0tw= 0.50 and  = 4.0 (see Fig. 2). The pulse time width twis related to trand  via tw= tr−−1( + 1) exp(), where (x) being the Euler gamma function. The radiated pulsed fields are observed within the bounded time window of observation T = {0 ≤ t ≤ 5.0 /c0}. Fig. 3. The -component of the radiated electric-type far-field characteristics. (a) Au wire segment; (b) PEC wire segment. In the first step, the -component of the radiated far-field characteristics is evaluated according to Eqs. (17) and (18) for {0 <  ≤ } and t ∈ T. Fig. 3a–b show the resulting normalized pulse shapes for the analyzed Au nanowire and its PEC equivalent, respectively. As expected, both wire segments do not radiate in the direction parallel to the wire axis ( = 0 and  = 180 (deg)). It has been observed that the peak of the pulsed field radiated from the PEC wire segment was relatively stronger with respect to the peak corresponding to the Au segment. While the former peak is observed at about  = 40 (deg), the Au nanowire shows its maximum radiation closer to the broadside direction ( = 90 (deg)). It is further observed that the separation of the positive and negative lobes of the signal radiated from the PEC wire segment gradually increases from  = 0 to  = 180 (deg) according to (/c0)[1 − cos()] (see Eq. (18)). This is not apparently the case for the Au segment whose radiated pulse shape is far more complex. An interesting property in this respect is the oscillatory behavior of the radiated signal from the Au segment in about {70 <  < 80} (deg) (see Fig. 3a). This behavior may be useful for designing compact secure digital transmission systems, where the drastic distortion of the signal in specific directions is desirable. In order to further illustrate some specific features of the pulsed EM radiation phenomena, three-dimensional TD radiation patterns have been plotted at two observation times t = {0.2 /c0, 0.6 /c0} ∈ T. Fig. 4a,b and c,d show the results for the Au and PEC nanowires, respectively. In particular, an inspection of the TD radiation patterns at t = 0.2 /c0(see Fig. 4a,c) shows that the main lobe of the PEC segment is oriented toward a lower elevation angle ( ≃ 28 (deg)) with respect to the one of the Au nanowire ( ≃ 75 (deg)). Finally, 38 M. ˇ Stumpf, G.A.E. Vandenbosch / Photonics and Nanostructures – Fundamentals and Applications 22 (2016) 35–39 Fig. 4. The three-dimensional radiation patterns of the plasmonic Au nanowire at (a) c0t/ =0.2; (b) c0t/ =0.6 and of the PEC nanowire at (c) c0t/ =0.2; (d) c0t/ =0.6. the distinct separation between the leading positive and negative lobes of the radiated signal from the PEC segment can be clearly seen in Fig. 4d, where the radiated field is plotted at t = 0.6 /c0. To gain a further insight into the simplified analytical model and its limitations a fully three-dimensional numerical model of the wire segment has been analyzed using the Finite Integration Technique (FIT) as implemented in CST Microwave Studio®. In this example we analyze a lossy Au plasmonic wire segment that is described by ωp= 8.55 (eV) and c= 0.0184 (eV) (see Eq. (1)). The conducting cylindrical antenna of a finite radius 1.0 (nm) and length =500 (nm) is excited via an electric-current ‘discrete port’ placed in between a thin circular screen of radius 2.0 (nm) and the wire. The length of the excitation gap between the antenna and the screen is 1.0 (nm). As the excitation electriccurrent pulse we take the one as shown in Fig. 2, again. Fig. 5(a) and (b) show the (normalized) TD far-field characteristics that correspond to the Au and PEC segments, respectively. At first, owing to the excitation port-wire coupling, one must expect relatively lower radiated amplitudes with respect to the ones offered by the idealized analytical model. For a mathematical analysis of the relevant excitation mechanisms, we refer the reader to [23, Section 16.11]. Secondly, an inspection of Figs. 3b and 5b clearly reveals the effect of a reflected TD current constituent on the far-field amplitude radiated from the PEC wire segment. As the travelingcurrent waves get attenuated along a lossy wire segment, one can expect that this effect will become weaker for non-perfectly conducting antenna structures met in practice. Moreover, since the (secondary) reflected-current constituents do not manifest themselves until the (primary) excitation electric-current pulse reaches the antenna end-points, the introduced analytical model will predict fairly accurate results in the early-time part of the response. It is finally noted in Fig. 5a that the distortion of the pulsed field radiated from the finite lossy Au wire segment is more uniform over {0 <  ≤ } than the one corresponding to the idealized model (see Fig. 3a). A more detailed analytical study of the wave excitation and propagation mechanisms is necessary to fully grasp such a complex space-time effect and its potentialities. Fig. 5. The -component of the radiated electric-type far-field characteristics of as evaluated numerically using FIT of CST Microwave Studio®. (a) Au wire segment; (b) PEC wire segment. 6. Conclusions Closed-form analytical formulas describing the pulsed EM field radiation from an electric-current pulse traveling along M. ˇ Stumpf, G.A.E. Vandenbosch / Photonics and Nanostructures – Fundamentals and Applications 22 (2016) 35–39 39 a plasmonic nanowire have been derived with the aid of the unilateral Laplace transformation. It has been demonstrated that the pulsed EM characteristics of such a radiator can be expressed as the sum of the characteristics of the corresponding PEC wire segment and the relaxation contribution describing its plasmonic behavior. 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