2116 J. Opt. Soc. Am. B / Vol. 25, No. 12 / December 2008 Talebi et al. Plasmonic ring resonator Nahid Talebi,* Ata Mahjoubfar, and Mahmoud Shahabadi Photonics Research Laboratory, Center of Excellence for Applied Electromagnetic Systems, School of Electrical and Computer Engineering, University of Tehran, North Kargar Ave., Tehran, Iran *Corresponding author:
[email protected]Received May 19, 2008; revised September 29, 2008; accepted October 3, 2008; posted October 8, 2008 (Doc. ID 96237); published November 26, 2008 A ring resonator composed of a plasmonic waveguide is presented. For the plasmonic waveguide, an array of silver nanorods is assumed. To determine the modes of this ring resonator, the generalized multipole technique (GMT) is used. Using this analysis, we obtain various modes of the proposed ring resonator. The mode field and the corresponding quality factor of each mode of the ring resonator is computed. The results are compared with those obtained using the finite-element method (FEM) and the finite-difference time domain (FDTD) method. © 2008 Optical Society of America OCIS codes: 250.5403, 240.6680, 350.4238, 230.5750. 1. INTRODUCTION method with exponential convergence when applied to smooth boundaries [15]. Using this method, one can esti- Plasmonic waveguides for efficient guiding of light below mate performance of plasmonic structures with high ac- the diffraction limit were first presented in [1] and later curacy. We have used the GMT codes developed in our investigated in [2–6]. A specific kind of these waveguides laboratory to analyze the proposed plasmonic ring resona- is comprised of an array of nanoparticles where the near- tor. field coupling of the closely spaced nanoparticles sets up coupled plasmon modes with features such as slow-wave propagation and negative group velocity [7]. These arrays have also been examined experimentally [8,9]. The ex- 2. ANALYSIS OF PLASMONIC RING perimental results agree with the numerical simulations RESONATORS USING GMT based on electromagnetic optics, which could also prove Figure 1 shows the structure of a plasmonic ring resona- that the tunneling of electrons between adjacent nanopar- tor and its unit cell. The structure is comprised of N na- ticles is negligible. However, there are still several chal- lenges regarding the design of components such as bends. y Dielectric micro-ring and disk resonators are key ele- n=3 ments in photonic integrated circuits and have been in- n=2 vestigated using methods like the finite-difference time Unit cell domain (FDTD) method [10], the coupled-mode theory [11], and the generalized multipole technique (GMT) [12]. These structures can be used for designing channel-drop rp filters and lasers [13]. Arranging them in arrays to realize slow-wave waveguides has also been reported [14]. Simi- r lar progress in designing nanoscaled circuits can be made Rrod by the introduction of plasmonic ring resonators based on x a plasmonic waveguide. This is the goal of the present work. We believe that such a ring resonator can be used D (ε ) D2(εr2) for the coupling of light into plasmonic waveguides and 1 r1 for realization of nanoscaled filters, to name a few. Here, our aim is to investigate the performance of a plasmonic L n=N ring resonator comprised of nanorods arranged in a circu- lar array. Different circulating and whispering gallery modes of this structure will be studied for their mode fields and quality factors. Computation of the eigen wave- n=N−1 lengths is carried out using the GMT. It should be men- Fig. 1. (Color online) Plasmonic ring resonator. The locations of tioned that many computational techniques are chal- the multipoles for expanding the fields in domains D1 and D2 are lenged by plasmonic structures since the electromagnetic represented by ⫻ and 쎻, respectively. The points marked by ⴱ field is highly enhanced on the boundaries of the nanopar- represent the excitation points. In this structure, Rrod = 25 nm, ticles. The GMT is a semi-analytical frequency-domain L = 75 nm, and N = 10 are assumed. 0740-3224/08/122116-7/$15.00 © 2008 Optical Society of America Talebi et al. Vol. 25, No. 12 / December 2008 / J. Opt. Soc. Am. B 2117 Fig. 2. (Color online) Comparison between the GMT and the FEM results. The left inset visualizes the magnitude of Hz component at = 284 nm computed using the GMT with the shown multipoles. The right inset is the configuration of the meshes for the FEM analysis. norods with a radius of Rrod = 25 nm and a center to center excitation magnetic fields in domain Dn at observation spacing of L = 75 nm. The resultant plasmonic ring reso- point r, respectively. In domain Dn, we can derive other nator is a highly symmetric configuration. In order to field components Ex and Ey from HD z 共r兲. The coordinate n compute the eigen frequencies of the resonator using the system used here is depicted in Fig. 1. GMT, a cyclic symmetry decomposition technique is used. To model the structure, we decompose the unit cell into This considerably reduces the simulation time and in- two subdomains. The scattered field in domains D1 and creases accuracy. In this technique we only compute the D2 is expanded using the multipoles located at the indi- fields within a unit cell of the structure. cated points in Fig. 1. Thus, the magnetic field can be The GMT is based on expansion of the field in terms of given by appropriate basis functions such as multipolar functions P1,2 N⬘ and the modes of waveguides. They are solutions to the Helmholtz equation. To apply this method to a two- D Hz,s1,2共r, 兲 = 兺 兺 共A p=1 n=1 1,2 n cos共np兲 + B1,2 n sin共np兲兲 dimensional (2D) region composed of multiple domains D1 D to Dn with constitutive parameters n and n, the field in ⫻exp共j共p − 1兲M0兲Bn 1,2共k1,2兩r − rp兩兲 , 共2兲 each domain is expanded in terms of analytical solutions of the 2D Helmholtz equation for the corresponding do- where the time dependence is assumed as exp共jt兲 in main. For a TEz incident field (which is the case for the which is the angular frequency. For expanding the excitation of plasmon resonances of nanorods), the only fields in domain D1 with the constitutive parameters r1 nonzero field components are Hz, Ex, and Ey. The total = 1 and r1 = 1, a corona of multipoles is used at the center 共2兲 magnetic field can be expressed as of all of the nanorods such that in Eq. (2) BD n = Hn is the 1 Hankel function of the n-th order and the second kind. D D D Hz n共r兲 = Hz,sn共r兲 + Hz,in共r兲 , 共1兲 The amplitude of each multipole cluster outside the unit cell is determined by the amplitude of the multipole clus- in which HzDn共r兲 is the total magnetic field in domain Dn. ter located in the unit cell multiplied by the factor Dn Dn Moreover, Hz,s 共r兲 and Hz,i 共r兲 denote the scattered and the exp共j共p − 1兲M0兲, where M is the unknown mode number of the ring resonator, p = 1 , 2 , . . . , 共P2 = N兲, and 0 = 2 / N is Table 1. Iterations of FEM Solution to Achieve the unit-cell angle. The field in domain D2 is expanded us- Convergence for the Transmittance at = 140 nm ing the Bessel functions of the first kind, BD n = Jn , whose 2 origin is at the center of the nanorod located in the unit Number of Number of Magnitude of Iteration triangular elements boundary elements magnetic fields cell. In Eq. (2), p represents the azimuth angle at which the field point r is seen by the p-th multipole located at 1 26100 1000 1.12927 rp. P1,2 is the total number of multipole clusters for ex- 2 38562 1340 1.12990 panding the fields in domains D1 and D2, and N⬘ denotes 3 57840 1788 1.13024 the maximum order of multipoles in each cluster. Here, 4 63238 1900 1.13033 k1,2 = 冑00r1,2, where r2 is the complex relative per- 5 71712 2116 1.13038 mittivity of the subdomain D2. 6 81400 2336 1.13042 For r2, we will assume the Drude model of silver [16]. 7 95308 2656 1.13044 We do not consider the damping rate resulting from sur- face damping effects or surface scattering-limited mean- 2118 J. Opt. Soc. Am. B / Vol. 25, No. 12 / December 2008 Talebi et al. Fig. 3. (Color online) Comparison between the GMT and the FDTD results. The left inset visualizes the Hz component at t = 14.2 fs computed using the FDTD method. The right inset shows Hz as a function of time at the observation point 共−R , 0兲. free-path effects, since it has been reported in the litera- To compute the eigen frequencies of the structure, one ture that these terms are negligible for nanoparticles can introduce a fictitious excitation at certain points in whose radius is greater than 10 nm [17]. In addition, we the unit cell and record the field at an observation point. have just considered the contribution of the s-band elec- In order to record different mode numbers of the system trons in the Drude model, and the influence of the d-band separately and with high accuracy, the excitation used is electrons or the interband transition has been neglected. made of N monopoles located on a circle, as shown in Fig. The contribution of the interband transitions in the 1. Their amplitudes are multiplied by exp共j共p − 1兲M0兲, Drude model is referred to as the size-dependent part of i.e., the same factor introduced for the corona of multi- the permittivity. For more details one is referred to [17]. poles. There are many ways to extract the plasmonic reso- For this work, the maximum order of each multipole nances of the metallic nanorods, such as computing the cluster and also the Bessel expansion is N⬘ = 9. Note that near-field enhancement or extinction cross section [18]. the chosen location and orders of the multipoles have pro- One may also use other search functions such as the duced highly convergent results with a maximum relative power flux, the energy in the unit cell, or the residue error of only 0.3% in satisfying the boundary conditions. search function. The residue search function is the re- Fig. 4. (Color online) Search functions for obtaining resonance wavelengths: a) N = 18, b) N = 19, c) N = 25, and d) N = 26. Talebi et al. Vol. 25, No. 12 / December 2008 / J. Opt. Soc. Am. B 2119 Table 2. Resonance Parameters of a Plasmonic Ring Resonator with L = 75 nm and Rrod = 25 nm Quality Resonance Quality Resonance factor for wavelength for factor for wavelength for N M L mode L mode (nm) T mode T mode (nm) 9 38.6 289.0 96.9 237.7 25 10 82.3 281.2 87.5 229.6 11 155.9 276.9 201.9 225.1 12 120.7 275.2 9.7 222.9 9 31.4 292.6 133.0 241.2 26 10 68.9 283.6 82.6 232.2 11 109.7 278.4 80.5 226.4 12 120.1 275.8 97.5 223.9 sidual of the error in satisfaction of the boundary condi- 1 / 冑2. The total number of time steps in our simulation is tions. This has been used in [7] and is shown to be effi- 4 ⫻ 104. A Ricker wavelet plane wave with a pick wave- cient in recording all the modes of a plasmonic waveguide. length of 284 nm and a unit amplitude for the magnetic Here, the residue search function is used to compute the field component Hz has been used as the incident wave on mode fields of the plasmonic ring resonator. the total-field-scattered-field (TFST) boundary. For the Alternatively, one may obtain the response of the struc- sake of comparison, the frequency-domain results are also ture to an appropriately chosen excitation. Here we con- shown in Fig. 3. It is apparent that while the results ob- sider two kinds of excitation: plane-wave excitation and a tained using GMT and FEM show two prominent reso- circulating wave located at the origin and given by nances at 278.8 nm and 286.2 nm, the FDTD results show just one resonance at 281 nm. We think that this is due to Hz,i = H共M2兲共k0r兲exp共jM兲 , 共3兲 the stair-casing error of FDTD at the curved boundaries of the rods. in which M is the mode number. To investigate the mode wavelengths and fields of the resonator, we have computed the eigen wavelengths of the structure after analyzing a unit cell as described previ- 3. NUMERICAL RESULTS ously. In Fig. 4, we have shown the residue search func- In order to verify our results, we calculated the transmit- tion for N = 18, 19, 25, and 26 for the mode numbers M tance and compared the obtained GMT results with those produced by both the FEM and the FDTD. Figure 2 de- picts the transmittance, which is the magnitude of the to- tal magnetic field component Hz at the point 共x , y兲 = 共−R , 0兲 when the resonator is illuminated by a TEz plane wave with a unit amplitude for its magnetic field compo- nent Hz and with the polarization depicted in the inset of Fig. 2. R denotes the radius of the plasmonic ring resona- tor. In the two insets, we have also shown both the loca- tions of the multipoles in the GMT and the configuration of the meshes used for the FEM analysis. All the multi- pole clusters have the same maximum order of N⬘ = 4. The results obtained using the GMT show excellent agree- ment with the ones obtained using the FEM. For the FEM analysis, a total number of 71712 triangular mesh ele- ments and 2116 boundary elements in an area of 1 m ⫻ 1 m have been used. On the outer boundaries of the simulation area, the scattering boundary condition is as- sumed. The mentioned number of elements leads to con- vergent results with a relative error of 8.85⫻ 10−3 per- cent. In order to demonstrate the convergence of the FEM results, the transmittance defined above has been com- puted for various mesh sizes and shown in Table 1 at the shortest wavelength of = 140 nm. For the FDTD analysis, we have considered a simula- tion domain of 1 m ⫻ 1 m, which has been discretized Fig. 5. (Color online) Magnitude of the magnetic field compo- by rectangular cells of ⌬x = ⌬y = 0.5 nm. For the bound- nent Hz for the L mode at a given time. The unit of the color bar aries of the simulation domain the uniaxial perfectly is A / m. (a) N = 25, M = 9, and = 289 nm; (b) N = 25, M = 10, and matched layers (UPML) has been used. The Courant = 281.2 nm; c) N = 25, M = 11, and = 276.9 nm; d) N = 25, M number 共c⌬t / ⌬x兲 has been set to its optimum value of = 12, and = 275.2 nm. 2120 J. Opt. Soc. Am. B / Vol. 25, No. 12 / December 2008 Talebi et al. Fig. 6. Computed electromagnetic power along the circumference depicted in the inset. = 7 , 8 , . . . , 12. The results depicted in Fig. 4 show three numbers. As expected, by decreasing the mode number, groups of modes. They belong to the longitudinal (L), the resonance wavelength of the ring resonator increases. transverse (T), and quadrupole (Q) plasmonic resonances, For both N = 25 and N = 26, increasing the mode number where the L and Q modes have the highest and lowest from M = 9 to M = 11 leads to higher quality factors, but resonance wavelengths, respectively. One can see that the the resonance mode with M = 12 has a lower quality factor free spectral range (FSR) for this resonator is very small. when compared with the resonance mode for M = 11. Two For the mode numbers M = 11 and M = 12, the FSR is only loss mechanisms are present in this structure, namely the 1.8 nm for the L mode. In Table 2, we have reported reso- radiation and the metallic losses. Obviously, they restrict nance parameters such as quality factor and resonance the quality factor of the plasmonic ring resonator. wavelength for different numbers of nanorods and mode To model the circulating modes of this structure, the ex- citation defined by Eq. (3) has been used to illuminate the structure. The scattered field has been expanded using the same multipoles as those used in deriving the results of Fig. 2. The scattered field of the plasmonic ring resona- tor has the same mode number as the M introduced in Eq. (2). Figure 5 shows the Hz component for the L modes of a plasmonic ring resonator with N = 25. This is computed for mode numbers M = 9 , . . . , 12. To verify our results, the resonance wavelength of the structure for M = 11 has been computed using the same excitation and integrating the Poynting vector along the circumference shown in the in- set of Fig. 6. The computed power is depicted in Fig. 6. The wavelength at which the real part of the power at- tains its maximum is the resonance wavelength for which the imaginary part of the power drops to zero. This wave- length coincides with the eigen wavelength computed us- ing the symmetry decomposition technique mentioned above and reported in Table 2. Note that the mode fields shown in Figs. 5(b) and 5(c) for the mode numbers of M = 10 and M = 11 are due to the propagation of two waves with the same mode number in the opposite directions of ˆ and −ˆ , where ˆ is the unit vector of the cylindrical coordinate system. Figure 5(d) shows the Hz component for M = 12, which is a whispering gallery mode of the system. This mode is especially inter- Fig. 7. (Color online) Magnitude of the magnetic field compo- nent Hz for the T mode at a given time. The unit of the color bar esting for the applications like the excitation of the whis- is A / m. (a) N = 25, M = 9, and = 237.7 nm; b) N = 25, M = 10, and pering gallery modes of an optical fiber. = 229.7 nm; c) N = 25, M = 11, and = 225.1 nm; d) N = 25, M Figure 7 shows the Hz component of transverse mode = 12, and = 223.3 nm. for N = 25 and the mode numbers M = 9, 10, 11, and 12. Talebi et al. Vol. 25, No. 12 / December 2008 / J. Opt. Soc. Am. B 2121 Table 3. Resonance Parameters for Various Plasmonic Ring Resonators with R = 299.2 nm, s = 25 nm, M = 12, and Different Values for Rrod Quality Resonance Quality Resonance factor for wavelength for factor for wavelength for N L mode L mode (nm) T mode T mode (nm) Rrod (nm) 26 156.2 269.4 207.8 221.4 23.6 27 154.0 264.4 217.5 219.8 22.2 28 147.5 259.9 182.7 218.6 21.0 29 146.0 255.8 200.2 217.4 19.8 30 143.8 252.0 213.2 216.4 18.8 Table 4. Resonance Parameters for Various Plasmonic Ring Resonators with R = 299.2 nm, Rrod = 25 nm, M = 12, and Different Values for L Quality Resonance Quality Resonance factor for wavelength for factor for wavelength for N L mode L mode (nm) T mode T mode (nm) L (nm) 26 152.4 275.2 191.1 224.9 72.1 27 140.8 275.7 174.5 226.9 69.5 28 126.8 276.5 135.9 229.0 67.0 29 105.5 277.5 92.7 231.5 64.7 30 80.6 278.6 58.6 234.5 62.6 The mode fields for the transverse mode have nearly the vestigated. For plane-wave excitation, the magnetic field same specifications as the longitudinal mode. The quality at an arbitrary field point has been computed using the factor for the T mode is higher than that of the L mode GMT, FEM, and FDTD methods. Excellent agreement has except for the mode number M = 12, which belongs to a been observed between the GMT and FEM results. Reso- whispering gallery mode. It is obvious from Fig. 7 that the nance wavelengths of several plasmonic ring resonators intensity of the Hz component inside the nanorods is with different radii have been computed to examine the greater for the L mode in comparison with the T mode. performance of the structure. The quality factors of the L In Tables 3 and 4, we have shown the parameters of the and T modes have been reported. The T modes have plasmonic ring resonator for different numbers of nano- higher quality factors than the L modes. Several mode rods and M = 12, while the radius of the ring resonator is fields have been obtained. Among these modes, there also kept unchanged. The results of Table 3 are for R exist whispering gallery modes. The proposed plasmonic = 299.2 nm, s = L − 2Rrod = 25 nm and different values for ring resonator has generally a low FSR. The dependence Rrod. As can be seen, by increasing the number of rods the of the quality factor on the radius of the nanorods and quality factor for the L mode decreases. Table 4 shows the their center to center spacing have been studied. It has results for R = 299.2 nm, Rrod = 25 nm, and different been shown that by increasing the number of nanorods center-to-center spacings. For the results shown in Table while keeping the radius of the resonator unchanged, the 4, the area of the metallic region is greater than the one quality factor for the L modes decreases. assumed in Table 2 with N = 25, so the rates of the reduction-of-quality factor for the L mode is higher. To compare the proposed plasmonic ring resonator with a bulk two-dimensional ring resonator, we investigate a ACKNOWLEDGMENTS two-dimensional ring resonator with a mean radius of R The authors thank Christian Hafner of the Swiss Federal = 299.2 nm, which is the same as the radius of the plas- Institute of Technology Zurich for his generous help with monic ring resonator reported in Table 2 with N = 25. The regard to the GMT method. resonance wavelengths of the two-dimensional ring reso- nator can be determined analytically. To obtain compa- rable resonance wavelengths, the width of the two- dimensional ring resonator is chosen to be 26.11 nm. For REFERENCES the bulk two-dimensional resonator, the resonance wave- 1. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Ausseneg, “Electromagnetic field transport via linear chains of silver length and the quality factor for the corresponding mode nanoparticles,” Opt. Lett. 23, 1331–1333 (1998). are = 275.9 nm and Q = 122.09, respectively. 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