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The radiation of a dipole emitter close to a metallic nanowire optical antenna is investigated theoretically. By considering the excitation and multiple scattering of surface plasmon polaritons (SPPs) on the antenna and neglecting all other surface waves, we build up an intuitive pure-SPP model to comprehensively describe the radiation of the antenna. The model shows that for antennas with short lengths that support lower orders of resonance, waves other than SPPs contribute considerably to the antenna radiation, while SPPs play a dominant role for other cases. The enhancement of the antenna radiation is shown arising from two contributions, the field directly radiated by the emitter and the field resonantly excited by the surface waves on the antenna.
© 2014 Optical Society of America
1. Introduction
Metalic nanostructures have a strong influence on the nearby emitters. Among various structures nanoantennas attract intense interests due to the ability of enhancing the fluorescence of molecules [1] or the spontaneous emission of quantum dots [2,3] and of directing the emission [4,5]. With the excellent properties in manipulating and controlling radiation of emitters, nanoantennas have potential applications in bio-chemical sensing [6], nano-devices of light emission [2,3], near-field spectroscopy [7] etc. To achieve an understanding of the underlying physics for guiding the design, analytical models have been built up to describe the near- and far-field properties of nanoantennas. The resonance frequency, quality factor and extinction cross-section of the nanowire antenna can be deduced analytically by modelling the antenna as a circuit composed of resistors, inductors and capacitors [8,9]. By treating standing waves of surface plasmon polaritons (SPPs) on the antenna as a superposition of two counter-propagating dipole currents, a current model is presented to explain the emission intensity of a nanowire antenna. The model however exhibits deviations from fully-vectorial numerical results [10]. The reflection coefficient of the SPP on a metallic nanowire is calculated with analytical formula [11]. The resonance wavelength is theoretically modelled by considering the amplitude of the surface plasma oscillations on the nanowire antenna [12]. A current model of nanowire antenna is built up by describing the antenna with homogeneous volume current instead of surface current [13]. An analytical model is presented by treating the nanowire antenna as a one-dimensional nano-cavity of SPPs [14]. With the current distribution on the antenna described with two counter-propagating SPPs, the far-field radiation pattern can be predicted with the model [14]. In previous literatures, it is commonly believed that the scattering or radiation properties of nanoantennas are governed by SPP modes that propagate on the antenna. However, this intuitive belief, which plays a central role in present antenna theories, has not been checked at a quantitative level until now. Even with this pure-SPP picture, the dynamical multiple-scattering process of SPPs on nanoantennas still requires clarification.
In this paper, we build up a pure-SPP model to analyze the radiation of an electric dipole emitter close to a nanoantenna composed of a single metallic nanowire. The model is derived by considering a dynamical process that the SPP is firstly excited by the emitter, then propagates along the antenna before further scattered at the end of the antenna. In the model we only consider the SPP and neglect all other waves on the antenna, so as to discriminate the respective contributions of the SPP and of other waves to the emission. The total and radiative emission rates as well as the far-field radiation pattern can be comprehensively predicted with the model. Analysis of the model shows that for antennas with large lengths that support higher-order resonances, the SPP imposes a dominant impact on the enhanced radiation of the emitter. However, for antennas with short lengths, the model shows that waves other than SPPs on the antenna also considerably contribute to the antenna radiation. All the conclusions are carefully supported through comparison between the model predictions and the fully-vectorial numerical results.
2. Fully-vectorial numerical results
As shown in Fig. 1(a), the nanoantenna considered here is composed of a metallic nanowire (length L) with a square cross section (side length D). The antenna is illuminated by a z-polarized electric dipole source (such as a molecule [1] or quantum dot [2,3]) placed at a distance d away from the antenna end. Here we use a normalized electric dipole source expressed as an electric current J = jzz = δ(x,y,z−zs)z, with δ the Dirac function, zs = L/2 + d, and z the unit vector along z direction. The antenna is made of gold put in air (refractive index nair = 1). For the calculation, we set the wavelength λ = 1μm, and the wavelength-dependent gold refractive index nm takes tabulated values from [15] (nm = 0.25 + 6.84i at λ = 1μm).
Fig. 1 (a) Geometry of the nanoantenna, which is illuminated by a normalized z-polarized electric dipole source (J) = jzz = δ(x,y,z−zs)z at the antenna end. (b)-(d) Scattering coefficients t and r, and scattered fields , and that appear in the SPP model. (e) Modulus of electric-field components of the fundamental SPP mode on the x-y cross section, which is calculated for D = 0.04λ, λ = 1μm (nm = 0.25 + 6.84i for gold).
We use an aperiodic-Fourier modal method (a-FMM) [16,17] to obtain the rigorous data for the antenna emission. The a-FMM is a generalized version of the well-developed rigorous coupled wave analysis (RCWA) [18,19]. RCWA has been widely used for solving the frequency-domain Maxwell’s equations of periodic systems upon Fourier basis. By incorporating perfectly matched layers to build up artificial periodicity, the a-FMM can then be applied to aperiodic structures with the same algorithm of the RCWA (some details about the a-FMM can be found in the Appendix).
With the a-FMM, we calculate the total emission rate Γtot and the radiative emission rate Γrad of the source. As shown in Figs. 2(a) and 2(b) (red circles), the Γtot and Γrad are plotted as functions of the antenna length L. For the calculation, Γtot = −Re[Ez(0,0,zs)]/2, where Ez(0,0,zs) is the z component of the electric field at the source position (x = 0, y = 0, z = zs). The Γrad is obtained through an integration of the normal component of the Poynting vector S on a closed surface encompassing the emitter and the antenna. Both Γtot and Γrad are normalized with the emission rate of the electric dipole source in the free space of air [20], Γair = η0k02nair/(12π), with η0 the wave impedance in vacuum, nair = 1 the air refractive index and k0 = 2π/λ.
Fig. 2 Total (a) and radiative (b) emission rates Γtot and Γrad (normalized with Γair in air), and quantum efficiency Γrad/Γtot (c) obtained for different antenna lengths L with the a-FMM (red circles) and the SPP model (blue solid curves). The horizontal dashed lines show the values for a semi-infinite nanoantenna. The results are obtained for d = 0.005λ, D = 0.04λ, λ = 1μm (for which t = 21.5232−36.7904i, r = −0.5721−0.6742i, neff = 1.4791 + 0.0346i in the SPP model).
As shown in Figs. 2(a) and 2(b) (red circles), Γtot and Γrad vary quasi-periodically and their peak values attenuate gradually with the increase of the antenna length L. Both Γtot and Γrad are drastically enhanced (Γtot>>Γair, Γrad>>Γair) for certain values of the antenna length. Compared with the case of the infinitely long antenna (L→∞, shown with the horizontal dashed lines in Fig. 2), which does not support surface-wave resonances due to the absence of the reflection of surface waves at the antenna end, Γtot for different L can be either higher or lower than the values for L→∞. This shows that the surface waves on the antenna launched by the emitter can either enhance or suppress the radiation of the source. However, for the radiative emission rate Γrad, its value is always higher than the value for L→∞. Figure 2(c) (red circles) provides the results for the quantum efficiency q = Γrad/Γtot [14], which characterizes the fluorescence quantum yield of emitters (such as molecules or quantum dots) with a high intrinsic quantum yield [21]. It is seen that with the increase of the antenna length L, the quantum efficiency q oscillates and peaks at the same resonance positions of Γtot and Γrad. Compared with the case of the infinitely long antenna (L→∞), q is always enhanced by surface waves on the antenna. Quantitative analysis will be provided later by building up surface-wave models.
The dependence of the total and radiative emission rates on the distance d between the source and the antenna end is investigated. As shown in Figs. 3(a)-3(c) (red circles) for antenna lengths L that correspond to the first three resonance peaks in Fig. 2, the emission rates Γtot and Γrad increase monotonously as the source-antenna distance d decreases. As d approaches zero (close to 1nm for instance), the Γtot increases rapidly while the Γrad approaches a constant, which implies an increase of the non-radiative emission rate (Γnonrad = Γtot − Γrad) that arises from the ohmic loss of the antenna. This causes a decrease of the quantum efficiency Γrad/Γtot, as confirmed by the experimental observation of the fluorescence quenching for nanometer-scale molecule-antenna distances [21].
Fig. 3 (a)-(c) Total (left) and radiative (right) emission rates plotted as functions of source-antenna distance d at the first three orders of resonance (m = 0, 1, 2). (d) The first two terms Γ1 and Γ2 in Eq. (4) of the SPP model and the SPP excitation coefficient t obtained for different d. The results are obtained for D = 0.04λ and λ = 1μm.
The angular distribution of the emitted power is calculated and the results for the first three resonance peaks of Γtot or Γrad are provided in Fig. 4.The angular emission pattern is obtained by calculating the modulus of the Poynting vector S on a circle around the antenna in the far-field region (the circle with a radius 8μm on the plane y = 0). As shown in Fig. 4(a), the emission patterns (red circles) exhibit one or more peaks along specific directions. With the increase of the order of resonances (or antenna lengths), the emission pattern successively exhibits two, four and six emission lobes. These results are consistent with earlier reports [14]. The emission pattern can be observed directly in Figs. 4(b), which show the x-z distribution (on the plane y = 0) of the modulus of the Poynting vector S obtained with the fully-vectorial a-FMM. Figures 4(d) show the x-z distribution of the near field on the antenna at the first three resonance peaks of emission rates. The results are obtained for electric-field intensities at the central cross section of the antenna (y = 0). The near field successively exhibits two, three and four intensity nodes for the first three orders of resonance. As shown hereafter with the model, the intensity nodes are induced by the interference of counter-propagating surface waves.
Fig. 4 (a) Angular distribution of the emitted power obtained at the first three orders of resonance (m = 0, 1, 2 from left to right). (b)-(c) Distribution of the Poynting vector modulus |(S)| (i.e. energy flux density) obtained for m = 0, 1, 2 with the fully-vectorial a-FMM and the SPP model. (d) Distribution of the electric near-field intensity on the cross section y = 0 for m = 0, 1, 2. The geometrical parameters of the antenna are the same as those in the Fig. 2. The international system of units (SI) is adopted for the presented data.
3. SPP model
To achieve a physical understanding of the numerical results, we build up an intuitive pure-SPP model. In the model only the SPP is considered and all other surface waves on the antenna are neglected, so that the impact of SPPs and other surface waves on the antenna radiation can be discriminated. Due to the deep sub-wavelength size of the antenna cross section, only the fundamental SPP mode is propagative and bounded at the antenna surface [22]. The electric-field amplitude of the fundamental SPP mode is shown in Figs. 1(e).
To build up the SPP model, we use a and b to denote the unknown coefficients of the down-going and up-going fundamental SPP modes on the antenna, as shown in Fig. 1(a). To determine the unknown a and b, a set of coupled-mode equations can be written,
In Eq. (1), t is the excitation coefficient of the down-going SPP by the dipole source [see Fig. 1(b)] [22], r is the reflection coefficient of the SPP at the top and the bottom ends of the antenna [Figs. 1(c) and 1(d)] [11], and u = exp(ik0neffL) is the phase shift of the SPP accumulated over one antenna length L, neff being the complex effective index of the SPP mode (k0 = 2π/λ). The scattering coefficients t and r along with the neff can be calculated with the fully-vecotrial a-FMM [16,17] which employs a stable scattering-matrix propagation algorithm [23] (some details about the calculation can be found in the Appendix). Equation (1a) is written in view that the down-going SPP originates from the excitation (coefficient t) of the dipole source and the reflection (r) of the up-going SPP (coefficient b) with a damping of u. Equation (1b) is written in a similar way. Solving Eqs. (1), we obtain the unknown a and b, Then we can obtain the radiated electromagnetic field of the antenna, for instance, the electric field being expressed as,In Eq. (3), E = [Ex, Ey, Ez] denotes the total electric-field vector, ES is the field of a semi-infinite nanoantenna illuminated by the source at the antenna end [see Fig. 1(b)], and the following two terms represent the scattered fields excited by the up-going and the down-going SPPs that propagate on the antenna. The is the scattered field for an incident up-going SPP mode on a semi-infinite antenna [Fig. 1(c)], and the is the scattered field for an incident down-going SPP [Fig. 1(d)]. For the calculation of and , the incident SPP that propagates on an infinite-length nanowire has been removed. With Eq. (3), the total emission rate can be expressed aswhere (0,0,zS = L/2 + d) is the coordinate of the source.4. Comparison between fully-vectorial numerical results and predictions of the SPP model
We will check the accuracy of the SPP model against fully-vectorial numerical results, so as to show the respective contributions of the SPP and of other waves to the antenna radiation. It is also possible to achieve a physical understanding with the analytical model that is based on an intuitive multiple-scattering picture of SPPs. We first calculate the total (Γtot) and radiative (Γrad) emission rates and the quantum efficiency (Γrad/Γtot) for different antenna lengths L with the model, as shown with the blue solid curves in Fig. 2. The Γtot is obtained with Eq. (4) and the Γrad is calculated with Eq. (3) via an integration of the Poynting vector S.
To achieve peak values of the SPP coefficients a and b, at which antenna resonance occurs with an enhancement of the radiation [for instance, Γtot and Γrad take peak values according to Eqs. (3) and (4)], the following resonance condition should be satisfied [11],
where Re(neff) refers to the real part of the SPP complex effective index, and arg(r) is the argument of the SPP reflection coefficient, m being an integer. Equation (5) is derived by minimizing the denominator of Eq. (2) in view that |rexp(ik0neffL)| is close to 1 (see their values in the caption of Fig. 2). It is seen from Eq. (5) that at resonance, the phase change accumulated by the SPP that propagates back and forth over one round on the antenna is multiples of 2π, which obviously results in a constructive interference of the multiple-scattered SPPs. In Fig. 2, the resonance peaks successively correspond to m = 0, 1, 2,… (the corresponding antenna lengths are L = 0.25λ, 0.58λ, 0.92λ,… for λ = 1μm).In Fig. 2, the horizontal dashed lines show the emission rates for a semi-infinite antenna (L→∞), which corresponds to the ES term in Eqs. (3) and (4). The contribution from the SPP to the emission is represented by the last two terms on the right side of Eqs. (3) and (4). As shown by the blue solid curves, the SPP model agrees well with the fully-vectorial results (red circles) for large antenna lengths (for instance, at the resonances of m = 1, 2, …). Thus for these cases SPPs contribute dominantly to the antenna radiation. For antennas with short lengths (for instance, at the resonance of m = 0), the model predictions show distinct deviation from the a-FMM results, thus surface waves other than SPPs also contribute considerably to the radiation. With the increase of the antenna length L, the quasi-periodic oscillation behavior of the emission rates Γtot and Γrad can be understood by noticing their dependence on L via the phase-shift factor u = exp(ik0neffL). Due to the propagation loss of the SPP mode [Im(neff) = 0.0346 for Fig. 2], the factor u approaches zero (i.e. the case of L→∞) with the increase of L, which explains the gradual decrease of the oscillation amplitude of the emission rates.
For fixed L and other geometrical parameters of the antenna, the quality factor Q for the resonance is defined by Q = −Re(ωc)/[2Im(ωc)], with the complex resonance frequency ωc determined by solving the complex pole of Eqs. (2), i.e. 1−(ru)2 = 0 with the frequency as the unknown. The following analysis of the Q factor with the Fabry-Perot model (such as the present SPP model) is quite classical, for instance, referring to the analysis for photonic crystal cavities [24].
Now we move to Fig. 3 that shows the dependence of emission rates on the source-antenna distance d. For the total emission rate Γtot, the first two terms (denoted by Γ1 and Γ2) on the right side of Eq. (4) are plotted as functions of distance d [Figs. 3(d1) and 3(d2)]. The sum of Γ1 and Γ2 is shown nearly equal to the total emission rate Γtot obtained with the model, and the contribution from the third term (scattering of down-going SPP) is found to be negligibly weak. So within the realm of the SPP model, the enhancement of the total emission rate is mainly due to the field directly radiated by the source (the Γ1 term) and the field excited by the up-going SPP (Γ2). For the second term Γ2, the coefficient b of the up-going SPP depends on the distance d via the SPP excitation coefficient t [see Eq. (2b)]. It is seen that with the decrease of d [Figs. 3(d1)-3(d3)], Γ1 increases rapidly while Γ2 and |t| increase monotonically and gradually approach a constant, which explains the rapid increase of Γtot. For the radiative emission rate Γrad, Eq. (3) of the model shows that Γrad originates from three contributions, the field directly radiated by the source (the first term) and the field excited by the up-going and the down-going SPPs (the last two terms).
With the SPP model [see Eq. (3)], the angular distribution of the emitted power [blue-solid curves in Fig. 4(a)] as well as the modulus of the Poynting vector [Figs. 4(c)] are reproduced. At the first order of resonance (m = 0), the predictions of the model show distinct deviations from the fully-vectorial a-FMM data [red circles in Figs. 4(a) and Figs. 4(b)], while both results exhibit similar profiles. This indicates that for antennas with short lengths that support lower-order resonances, surface waves other than SPPs on the antenna contribute considerably to the radiation, consistent with the comparison shown in Figs. 2 and 3 for m = 0. At higher orders of resonance (m = 1, 2), the SPP model shows good agreement with a-FMM results. Thus for antennas of large lengths that support higher-order resonances, only the SPP has dominant impact on the radiation.
5. Residual waves on the antenna
The deviation of the predictions of the SPP model from the fully-vectorial a-FMM results show that there should exist waves other than SPPs that contribute to the antenna radiation, and their contribution is less pronounced for antennas with larger lengths. To see this directly, we calculate the residual field by removing the SPPs from the total field on the antenna. The total electromagnetic field and its contained SPPs are calculated with the fully-vectorial a-FMM [16,17] (see the Appendix for some details about the calculation). Figure 5 shows the tangential electric component of the SPP field (blue solid curves) and the residual field (red dashed curves) on the surface of the antenna. It is seen that at the first-order resonance with a short length of antenna [m = 0, shown in Fig. 5(a)], the residual field is appreciable compared with the SPP field. Whereas, at the higher-order resonances [m = 1, 2, shown in Figs. 5(b) and 5(c)], the residual field is weak compared with the SPP field. This observation evidences the contribution of the residual field to the antenna resonance, and is consistent with the higher accuracy of the SPP model at higher-order resonances. For a flat metal-dielectric interface, there have been works showing that besides the SPP, there exist other surface waves responsible for the electromagnetic interactions between scatterers on the interface, which are so-called quasi-cylindrical waves [25–32]. But for nanoantennas, the role of surface waves other than SPPs is rarely studied [8–14].
Fig. 5 Tangential electric-field component (|Ez| in SI) of the SPP field (blue solid curves) and of the residual field (red dashed curves) on the surface of the antenna (obtained at y = 0) at the first three orders of resonance (m = 0, 1, 2). The geometrical parameters of the antenna are the same as those in the Fig. 2.
6. Conclusion
We propose a simple pure-SPP model to investigate the radiation of an emitter close to a metallic nanowire optical antenna. In the model we only consider the SPP and neglect all other waves on the antenna, so as to discriminate the respective contributions of the SPP and of other waves to the emission. The model is built up by considering the dynamical excitation and multiple-scattering process of SPPs on the antenna. The features of the antenna radiation (such as the emission rates, angular emission pattern, field distribution and so on) can be comprehensively reproduced. The model is self-sufficient and does not rely on any fitting process with the use of fully-vectorial or experimental data. Our results show that the enhancement of the emission rates arises from the resonance of SPPs and other waves on the antenna. Comparison between the prediction of the SPP model and the fully-vectorial numerical results shows that for antennas with large lengths that support higher-order resonances, the SPP imposes a dominant impact on the radiation of the emitter. Whereas, for antennas with short lengths that support lower-order resonances, the SPP is not the unique wave on the antenna that contributes to the radiation, and the contribution of surface waves other than SPPs should be considered. This conclusion is further confirmed by a direct observation of the residual waves on the antenna. From the model we derive a resonance condition which is related to a constructive interference of the multiple-scattered SPPs on the antenna. Analysis of the model shows that the enhancement of the antenna radiation is mainly attributed to the field directly radiated by the emitter and the field excited by the surface waves that are resonantly excited and that propagate on the antenna. Our work can be helpful for clarifying the underlying physics of the radiation of optical nanoantenna, and for an intuitive design of the relevant devices. It is possible to extend the present model to more complex or practical antenna geometries, such as antenna arrays [33], Yagi-Uda antennas [4,5] and so on.
Appendix
Here we aim at providing some details on the fully-vectorial a-FMM [16,17] that we used to calculate the electromagnetic field (such as the emission rates and the far-field radiation pattern) and to obtain the quantities for building up the SPP model (such as the SPP mode and its scattering coefficients t and r).
In the a-FMM, perfectly matched layers satisfying outgoing-wave conditions are introduced in the transversal x- and y-directions as a complex coordinate transform [16], x=Fx(x') and y=Fy(y'), which map the infinite transversal range into a finite supercell, x'∈[−Λx/2,Λx/2] and y'∈[−Λy/2,Λy/2]. Then by building up an artificial periodic array (with periods Λx and Λy) of the supercell with respect to the new coordinates x' and y', the Maxwell’s equations are then reformulated to be periodic systems and can be solved with the algorithm of RCWA [18,19].
In RCWA, any component of the electromagnetic field is expanded upon the Fourier basis with respect to the periodic transversal coordinates x' and y'. With the expansion inserted into Maxwell’s equations, the resultant system of linear ordinary differential equations for the unknown Fourier coefficients (dependent on z) is then integrated analytically in each z-invariant layer, which expresses electromagnetic field in terms of waveguide eigenmodes [18,19]. For instance, the Ex component of electric field in each z-invariant layer can be expressed as
In Eqs. (6), M and N are the truncated harmonic numbers of Fourier series, Kx = 2π/Λx and Ky = 2π/Λy. βp is the eigenvalue of the coefficient matrix of the linear ordinary differential equations derived from the RCWA, and corresponds to the propagation constant of the waveguide eigenmodes. wm,n,p is the element of the corresponding eigenvector, and describes the transversal field distribution of the waveguide eigenmodes. cp+ and cp− are the coefficients of the up-going and down-going waveguide eigenmodes, and are determined by matching the continuity boundary condition of tangential electromagnetic field at the interfaces between adjacent z-invariant layers. For the antenna shown in Fig. 1(a) for instance, the whole space is divided into four z-invariant layers, the top free space, the free space between the source and the antenna end, the antenna layer and the bottom free space, which are separated by three horizontal interfaces. The source is treated as a discontinuity of tangential components of electromagnetic field. A stable scattering-matrix propagation algorithm (such as those summarized in Ref [23].) is adopted for the matching of the boundary condition.To obtain the quantities for building up the SPP model, the SPP mode is solved as one waveguide eigenmode in the z-invariant antenna layer, which corresponds to a specific term in Eq. (6b) (with βp=k0neff). Then the scattering coefficients t and r of the SPP mode can be obtained from the corresponding cp+ or cp− by matching the boundary condition. For the calculation of the residual field in Section 5, the total electromagnetic field in the antenna layer is obtained from Eqs. (6), and the residual field is then obtained by removing the SPP terms on the right side of Eq. (6b) (there exist two counter-propagating SPPs on the antenna).
Acknowledgments
This work is supported financially by the National Key Basic Research Program of China (973 Program) under Grant No. 2013CB328701, the Natural Science Foundation of China (NSFC) under Grant No. 61322508, and the Natural Science Foundation of Tianjin under Grant No. 11JCZDJC15400.
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