## Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation

Optics Express, Vol. 17, Issue 16, pp. 14211-14228 (2009)

http://dx.doi.org/10.1364/OE.17.014211

Acrobat PDF (1746 KB)

### Abstract

In this study, we report the investigation of both near- and far-field electromagnetic characteristics of two-dimensional silver nanorod arrays embedded in anodic aluminum oxide with the use of a high-accuracy three-dimensional Legendre pseudospectral time-domain scheme. The simulated far-field scattering spectra agree with the experimental observations. We show that enhanced electric field is created between adjacent nanorods and, most importantly, far-field scattered light wave is mainly contributed from surface magnetic field, instead of the surface enhanced electric field. The identified near-field to far-field connection produces an important implication in the development of efficient surface-enhanced Raman scattering substrates.

© 2009 Optical Society of America

## 1. Introduction

1. S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. **98**, 011101–011110 (2005). [CrossRef]

2. H. H. Wang, C. Y. Liu, S. B. Wu, N. W. Liu, C. Y. Peng, T. H. Chan, C. F. Hsu, J. K. Wang, and Y. L. Wang, “Highly Raman-enhancing substrates based on silver nanoparticle arrays with tunable sub-10 nm gaps,” Adv. Mater. **18**, 491–495 (2006), and references therein. [CrossRef]

3. T.-T. Liu, Y.-H. Lin, C.-S. Hung, T.-J. Liu, Y. Chen, Y.-C. Huang, T.-H. Tsai, H.-H. Wang, D.-W. Wang, J.-K. Wang, Y.-L. Wang, and C.-H. Lin, “A high speed detection platform based on surface-enhanced Raman scattering for monitoring antibiotic-induced chemical changes in bacteria cell wall,” PloS One (in press).

4. S. Biring, H.-H. Wang, J.-K. Wang, and Y.-L. Wang, “Light scattering from 2D arrays of monodispersed Ag nanoparticles separated by tunable nano-gaps: spectral evolution and analytical analysis of plasmonic coupling,” Opt. Express **16**, 15312–15324 (2008). [CrossRef] [PubMed]

5. Y. C. Chang, J. Y. Chu, T. J. Wang, M. W. Lin, J. T. Yeh, and J.-K. Wang, “Fourier analysis of surface lasmon waves launched from single nanohole and nanohole arrays: unraveling tip-induced effects,” Opt. Express **16**, 740–747 (2008). [CrossRef] [PubMed]

6. J. Zhao, A. O. Pinchuk, J. M. McMahon, A. Li, L. K. Ausman, A. L. Atkinson, and G. C. Schatz, “Methods for describing the electromagnetic properties of silver and gold Nanoparticles,” Acc. Chem. Res. **41**, 1710–1720 (2008). [CrossRef] [PubMed]

6. J. Zhao, A. O. Pinchuk, J. M. McMahon, A. Li, L. K. Ausman, A. L. Atkinson, and G. C. Schatz, “Methods for describing the electromagnetic properties of silver and gold Nanoparticles,” Acc. Chem. Res. **41**, 1710–1720 (2008). [CrossRef] [PubMed]

7. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**, 1491–1499 (1994). [CrossRef]

9. G. Mie, “Contributions to the optics of turbid media, especially colloidal metal solutions,” Ann. Phys. **25**, 377–445, (1908). [CrossRef]

6. J. Zhao, A. O. Pinchuk, J. M. McMahon, A. Li, L. K. Ausman, A. L. Atkinson, and G. C. Schatz, “Methods for describing the electromagnetic properties of silver and gold Nanoparticles,” Acc. Chem. Res. **41**, 1710–1720 (2008). [CrossRef] [PubMed]

10. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**, 302–307 (1966). [CrossRef]

11. S. S. Zivanovic, K. S. Yee, and K. K. Mei, “A subgridding method for the time-domain finite-difference method to solve Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. **39**, 471–479 (1991). [CrossRef]

13. T. Yamaguchi and T. Hinata, “Optical near-field analysis of spherical metals: application of the FDTD method combined with the ADE method,” Opt. Express **15**, 11481–11491 (2007). [CrossRef] [PubMed]

2. H. H. Wang, C. Y. Liu, S. B. Wu, N. W. Liu, C. Y. Peng, T. H. Chan, C. F. Hsu, J. K. Wang, and Y. L. Wang, “Highly Raman-enhancing substrates based on silver nanoparticle arrays with tunable sub-10 nm gaps,” Adv. Mater. **18**, 491–495 (2006), and references therein. [CrossRef]

14. R. M. Stöckle, Y. D. Suh, V. Deckert, and R. Zenobi, “Nanoscale chemical analysis by tip-enhanced Raman spectroscopy,” Chem. Phys. Lett. **318**, 131–136 (2000). [CrossRef]

15. E. Fort and S. Grésillon, “Surface enhanced fluorescence,” J. Phys. D **41**, 013001:1–31 (2008). [CrossRef]

16. X. Ji, W. Cai, and P. Zhang, “High-order DGTD methods for dispersive Maxwell’s equations and modelling of silver nanowire couping,” Int. J. Numer. Meth. Engng. **69**, 308–325 (2007). [CrossRef]

17. C.-H. Teng, B.-Y. Lin, H.-C. Chang, H.-C. Hsu, C.-N. Lin, and K.-A. Feng, “A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell’s Equations,” J. Sci. Comput. **36**, 351–390 (2008). [CrossRef]

2. H. H. Wang, C. Y. Liu, S. B. Wu, N. W. Liu, C. Y. Peng, T. H. Chan, C. F. Hsu, J. K. Wang, and Y. L. Wang, “Highly Raman-enhancing substrates based on silver nanoparticle arrays with tunable sub-10 nm gaps,” Adv. Mater. **18**, 491–495 (2006), and references therein. [CrossRef]

4. S. Biring, H.-H. Wang, J.-K. Wang, and Y.-L. Wang, “Light scattering from 2D arrays of monodispersed Ag nanoparticles separated by tunable nano-gaps: spectral evolution and analytical analysis of plasmonic coupling,” Opt. Express **16**, 15312–15324 (2008). [CrossRef] [PubMed]

## 2. Numerical scheme and calculation geometry

4. S. Biring, H.-H. Wang, J.-K. Wang, and Y.-L. Wang, “Light scattering from 2D arrays of monodispersed Ag nanoparticles separated by tunable nano-gaps: spectral evolution and analytical analysis of plasmonic coupling,” Opt. Express **16**, 15312–15324 (2008). [CrossRef] [PubMed]

*D*and a length of

*L*are arranged in a hexagonal pattern with an inter-nanorod gap of

*W*, and are embedded inside alumina matrix with only the half-sphere round top exposed in vacuum, as shown in Fig. 1(c). Although there exist some inhomogeneous variations in

*D*,

*L*,

*W*as well as the proportion of the rod exposed in vacuum [2

**18**, 491–495 (2006), and references therein. [CrossRef]

**16**, 15312–15324 (2008). [CrossRef] [PubMed]

*D*and

*L*were set to be 25 and 100 nm, respectively, and five different gaps (5, 10, 15, 20, 25 nm) were chosen to simulate the experimental condition. To fully incorporate the dispersive property of Ag over the spectral range of interest ranging from 400 to 800 nm into the adopted numerical model, its dielectric function is represented as a Drude-Lorentz model expressed by

*ε*

_{∞}is the high-frequency limit of the dielectric constant

*ω*and

_{p}*τ*are the plasma frequency and the life time in the Drude model of metal, respectively,

_{D}*ν*is the number of Lorentz oscillators with a natural frequency of

*ω*, a strength of

_{s}*f*, and a damping constant of Γ

_{s}_{s}in the Lorentz model to account for possible interband transitions in metal. Two Lorentz terms are adopted to accurately represent the experimental data taken from the work by Lynch and Hunter [18] during the parameter extraction process. Its detailed consideration is delineated in the Appendix. The dielectric constant of alumina was set to be 3 over the spectral range of interest [4

**16**, 15312–15324 (2008). [CrossRef] [PubMed]

19. D. W. Thompson, “Optical characterization of porous alumina from vacuum ultraviolet to midinfrared,” J. Appl. Phys. **97**, 113511:1–9 (2005). [CrossRef]

20. J. L. Young and R. O. Nelson, “A summary and systematic analysis of FDTD algorithms for linearly dispersive media,” IEEE Antennas Propag. Mag. **43**, 61–77 (2001). [CrossRef]

_{s}is the polarization corresponding to the

*s*-th Lorentz term, and J⃗ is the electric current density within the metal medium.

17. C.-H. Teng, B.-Y. Lin, H.-C. Chang, H.-C. Hsu, C.-N. Lin, and K.-A. Feng, “A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell’s Equations,” J. Sci. Comput. **36**, 351–390 (2008). [CrossRef]

*W*= 25 nm in Fig. 2, was first decomposed into a series of non-overlapping, curved hexahedral sub-domains. These sub-domain blocks in

*x*-space were then mapped one-to-one into a unit cube in

*ξ*-space. Secondly, a collocation method based on Legendre-Gauss-Lobatto (LGL) quadrature points was used to construct Legendre-Lagrange interpolation polynomials to accurately evaluate the field and its derivatives at these sampling points of the unit cube. The number of the sampling points on each edge of individual sub-domain,

*N*, is 15 and the resultant mesh grids in x-space in the array of

_{S}*W*= 25 nm are shown in Fig. 2. Thirdly, a low-storage fourth-order five-stage Runge-Kutta algorithm [21] was applied in temporal discretization for efficient memory use, as compared with the standard fourth-order Runge-Kutta scheme. Moreover, time-explicit boundary conditions in treating the input external electromagnetic wave at the Runge-Kutta intermediate stages [22

22. M. H. Carpenter, D. Gottlieb, S. Abarbanel, and W. S. Don, “The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: A careful study of the boundary error,” SIAM J. Sci. Comp. **16**, 1241–1252 (1995). [CrossRef]

17. C.-H. Teng, B.-Y. Lin, H.-C. Chang, H.-C. Hsu, C.-N. Lin, and K.-A. Feng, “A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell’s Equations,” J. Sci. Comput. **36**, 351–390 (2008). [CrossRef]

23. S. Abarbenel and D. Gottlieb, “On the construction and analysis of absorbing layer in CEM,” Appl. Numer. Math. **27**, 331–340 (1998). [CrossRef]

24. S. Abarbenel, D. Gottlieb, and J. S. Hesthaven, “Well-posed perfectly matched layers for advective acoustics,” J. Comput. Phys. **154**, 266–283 (1999). [CrossRef]

*χ*is a free parameter to stabilize the scheme. Q

_{r,s}=

*∂*P

_{r,s}/

*∂t*, where

*r*=

*x*,

*y*and

*z*. Furthermore,

*M*

^{(1)}

_{ijk}= diag[

*ε*

_{∞},

*ε*

_{∞},

*ε*

_{∞},

*μ*

_{0},

*μ*

_{0},

*μ*

_{0}], where

*μ*

_{0}is permeability of vacuum.

*M*

^{(2)}

_{ijk}= I

_{15×15}and

*δ*is Kronecker delta function.

_{ij}_{n×m}and 0

_{n×m}denote identity and zero matrices with

*n*rows and

*m*columns, respectively. [*]

_{n×m}=diag[*,*,*], where * is any element in {

*ω*

^{2}

_{p},

*ω*

^{2}

_{s},

*f*

_{s}*ω*

^{2}

_{p},

*τ*

^{-1}

_{D}, Γ

_{s}}. Note that the boundary conditions to be imposed on the boundary surfaces of each sub-domain are absorbed in

*P*of Eq. (3a), called the penalized terms that serve mainly to exchange the information between adjacent boundary surfaces of surrounding sub-domains and to stabilize the numerical scheme. The explicit expressions of the penalized terms for different types of boundary conditions and the proper value of the penalty parameter

_{ijk}*χ*for stable computations can be founded in Ref. 17

**36**, 351–390 (2008). [CrossRef]

**16**, 15312–15324 (2008). [CrossRef] [PubMed]

*θ*= 45°. According to the reciprocity theorem of electromagnetics [25], the calculated scattered spectrum is identical to that obtained in the experimental geometry. Furthermore, the electric field strength of the incident wave was set to be one for simplicity. Figure 3 depicts the incident wave and the scattering collection geometry used in the calculation over an array of Ag nanorods that is embedded in the AAO substrate and is arranged in the hexagonal pattern. The

*x*-axis is along the direction in which two adjacent nanorods are closest. As described above, to avoid simulating the whole array, a hexagonal unit cell with periodic boundary conditions (PBCs), as shown in Figs. 1(b) and (c), was used to save calculation time, assuming that the excitation source is a plane wave.

*σ*, was realized with the use of the near-to-far-field transformation developed based on the field equivalence principle [26

_{SC}26. S. R. Rengarajan and Y. Rahmat-Samii, “The field equivalence principle illustration of the establishment of the non-intuitive null fields,” IEEE Antennas Propagat. Mag. , **42**, 122–128 (2000). [CrossRef]

*E*⃗ and

*H*⃗ on the top surface of the Ag nanorod array, shown in Fig. 1(c), in frequency domain were calculated by performing Fourier transform of E⃗ and H⃗ in time domain, separately. The corresponding electric and magnetic current densities (denoted by

*J*⃗ and

*M*⃗, respectively) in frequency domain on the top surface of the array, Fig. 1(c), were subsequently calculated with the following two relations:

*n*̂ is the normal vector on the top surface. Through Green’s functions, the electric and magnetic vector potentials were then obtained to produce far-field components. In the numerical calculation of

*σ*, we followed its notation used in Ref. 7

_{SC}7. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**, 1491–1499 (1994). [CrossRef]

*σ*can be expressed as

_{SC}*P*and

_{IN}*P*are the power densities of the incident and scattered waves, respectively,

_{SC}*k*is the wavenumber in free space,

*S*is the top surface of the unit cell, as shown in Fig. 1(c),

*r*̂ = (sin

*θ*cos

*ϕ*,sin

*θ*sin

*ϕ*,cos

*θ*),

*r*⃗

_{0}stands for the source point vector with respect to the origin within the unit cell, and

*η*

_{0}is the intrinsic impedance of vacuum. The array factor for a hexagonal periodic pattern is expressed as

*A*denotes the amplitude contribution for each unit cell, as all the unit cells are identical.

*a*(=

*D*+

*W*) is the inter-nanorod spacing. As the illuminating area on the sample in the scattering experiments was approximately a square covering about 1

*μ*m

^{2}, the number of nanorods was estimated accordingly for each array. Finally, the total scattering field was obtained by integrating over

*ϕ*along the circular path in Fig. 3 to simulate the far-field scattering measurement configuration.

9. G. Mie, “Contributions to the optics of turbid media, especially colloidal metal solutions,” Ann. Phys. **25**, 377–445, (1908). [CrossRef]

*σ*is given below. The diameter of the sphere was set at 25 nm and the wavelength of the incident wave was chosen to be

_{SC}*λ*(= 0.47

_{R}*μ*m) at which the scattering intensity reaches maximum. Figure 4 shows the calculated

*σ*and the difference between the exact result from the Mie theory and the calculated value, ∣ Δ

_{SC}*σ*∣, as a function of the observation angle,

_{SC}*θ*. Notice that ∣Δ

*σ*∣ for

_{SC}*N*= 15 is less than 10

_{s}^{-6}. Furthermore, the maximal error of the field components on the surface of the sphere is less than 3×10

^{-3}, which corresponds to a maximal energy error of 10

^{-5}. This result is much smaller than the corresponding one obtained with the FDTD method [13

13. T. Yamaguchi and T. Hinata, “Optical near-field analysis of spherical metals: application of the FDTD method combined with the ADE method,” Opt. Express **15**, 11481–11491 (2007). [CrossRef] [PubMed]

*E*∣ and ∣Δ

*H*∣, occur at

*λ*reach about 10%, while the differences in percentage decrease dramatically as the wavelength moves away from

_{R}*λ*. On the other hand, the differences between the actual dielectric function of silver and the fitted value of both the real and imaginary parts at

_{R}*λ*are only about 2.5% (Appendix A), indicating that plasmonic resonance magnifies the introduced error of the dielectric function in the calculation. This comparison study illustrates the importance of the accuracy of the dielectric functions of metal used in the near-field calculation of electromagnetic properties of plasmonic structures.

_{R}## 3. Results and discussion

*x*- and

*y*-polarized incident light waves.

*I*represents the far-field scattering intensity normalized to its maximal value. Notice that the resonance peak wavelength increases and the resonance width is broadened as the gap decreases, which is qualitatively consistent with our experimental observation published previously [4

_{SC}**16**, 15312–15324 (2008). [CrossRef] [PubMed]

26. S. R. Rengarajan and Y. Rahmat-Samii, “The field equivalence principle illustration of the establishment of the non-intuitive null fields,” IEEE Antennas Propagat. Mag. , **42**, 122–128 (2000). [CrossRef]

*J*⃗ and

*M*⃗ on the top surface of the Ag nanorod arrays are the sources in the calculation of distant scattering fields. Figures 7(a) and (b) show the scattering intensity of each Cartesian component of the two surface current densities for the Ag nanorod array with

*W*= 25 nm in the

*x*-polarized excitation scheme, where the polarization of the incident optical wave is parallel with the

*x*-axis. Notice that the total scattering intensity in far field is mainly contributed by

*J*and

_{x}*M*and, furthermore, the contribution by

_{y}*J*is several times of that by

_{x}*M*. In contrast,

_{y}*J*and

_{y}*M*play the dominant role in the

_{x}*y*-polarized excitation scheme and the contribution by

*J*is similarly larger than that by

_{y}*M*as shown in Figs. 7(c) and (d). For the cases of other gaps, the same components also consistently dominate the far-field scattering intensity spectra. One question then emerges naturally: does this behavior only exist in the 45°-scattering geometry? The calculations of the scattering angles at 0° and 90° were performed similarly and show the same behavior, indicating that this surface electric current dominated character is genuine in nature for these Ag nanorod arrays.

_{x}*J*⃗ and

*M*⃗ were calculated by surface magnetic and electric fields according to Eq. (4), respectively, it is thus of great help to discuss the behaviors of the two surface fields to understand the reason why

*J*⃗, instead of

*M*⃗, dominates the far-field scattering. Figures 8(a) and (b) show the surface ∣

*E*⃗∣ and ∣

*H*⃗∣ at 458 nm, respectively, that give the maximal scattering intensity in the case of

*W*= 25 nm in the

*x*-polarized excitation scheme. Notice that the surface ∣

*E*⃗∣ is localized at the two ends of the Ag nanorods along the

*x*-axis, which is consistent with the expected local electric field distribution of an isolated dipole-like Ag nanorod at electromagnetic resonance [28

28. M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser & Photon. Rev. **2**, 136–159 (2008). [CrossRef]

*H*⃗∣ is delocalized over the whole surface and produces the corresponding

*J*⃗ that acts as the dominant source of the far-field scattering of our Ag nanorod arrays. The distribution of the surface ∣

*H*⃗∣ can be similarly observed in our calculation of Mie scattering described above. The two observations above thus provide the first important conclusion resulting from this study: the surface magnetic field of these Ag nanorod arrays produces the near-field current source that dominates the far-field scattering. In comparison, Figs. 8(c) and (d) show the surface ∣

*E*⃗∣ and ∣

*H*⃗∣ at the same wavelength, respectively, in the case of

*W*= 25 nm in the

*y*-polarized excitation scheme. The surface ∣

*E*∣⃗ is localized at the two ends of the Ag nanorods along the

*y*-axis and the surface ∣

*H*⃗∣ is also delocalized over the whole surface. As the gap decreases, the plasmonic coupling between adjacent Ag nanorods is enhanced as in the case of the

*x*-polarized excitation scheme. As an example, Figs. 9(a) and (b) show the surface ∣

*E*⃗∣ and ∣

*H*⃗∣ at 626 nm, respectively, that gives the maximal scattering intensity in the case of

*W*= 5 nm in the

*x*-polarized excitation scheme. Notice that the surface ∣

*E*⃗∣ extends over the region between adjacent Ag nanorods along the

*x*-axis, creating so-called “hot spots.” This local enhanced electric field was similarly obtained previously in nanoparticle dimers [29

29. E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimmers,” J. Chem. Phys. **120**, 357–366 (2004). [CrossRef] [PubMed]

*H*⃗∣ in this case spreads even more uniform over the whole surface than that in the case of

*W*= 25 nm. In comparison, Figs. 9(c) and (d) show the surface ∣

*E*⃗∣ and ∣

*H*∣ at 629 nm, respectively, in the case of

*W*= 5 nm in the

*y*-polarized excitation scheme. The surface ∣

*E*⃗∣ is mainly localized in the gap region along the two directions that are 30° with respect to the

*y*-axis, because the electric field component of the incident wave along the two directions is the largest. On the other hand, the surface ∣

*H*⃗∣ almost evenly covers the whole surface, just as the case in the x-polarized excitation scheme.

*E*⃗∣ and ∣

*H*⃗∣ illustrated above for all the Ag nanorod arrays can be comprehended on the basis of quasi-static dipole model [4

**16**, 15312–15324 (2008). [CrossRef] [PubMed]

*e*⃗. The concurrent magnetic field, on the other hand, surrounds the surface of the Ag nanorod, owing to the induced linear oscillatory electron motion along

*e*⃗. In the case of an isolated Ag nanorod dimer with a very small gap and the dimer axis is parallel with

*e*⃗, plasmonic coupling taking place between them synchronizes the motions of the free electrons inside the two individual nanorods, extending the localized electric field,

*per se*, into the gap region and simultaneously aligning the oscillatory electrons in the two separate nanorods. The two in-phase oscillatory dipoles similarly expand the magnetic field over the region enclosing the dimer. Finally, by the same token, the plasmonic coupling in the case of two-dimensional Ag nanorod arrays brings all the dipole oscillations in phase and thus eventually makes the magnetic field spread almost uniformly over the whole surface. The closer the inter-nanorod gap is, the more uniform the surface magnetic field is over the top surface. In comparison, despite the successful interpretation of the far-field scattering properties in our previous work [4

**16**, 15312–15324 (2008). [CrossRef] [PubMed]

*E*⃗∣ and ∣

*H*⃗∣, respectively, and

*S*represents the top surface region. Notice that the energy density corresponding to the surface ∣

*H*⃗∣ is significantly larger than the one corresponding to the surface ∣

*E*⃗. This fact is thus consistent with the observation that the surface electric current dominates the far-field scattering intensity, as the scattering intensity in far zone directly reflects the field strength of the surface electric and magnetic fields. Furthermore, both the average surface electric and magnetic energy densities decrease with the decrease in the inter-nanorod gap. This behavior agrees with the predicted behavior of the

*Q*factor of the arrays based on the quasi-static dipole coupling model published previously [4

**16**, 15312–15324 (2008). [CrossRef] [PubMed]

*J*⃗ and

*M*⃗ act as the virtual current sources on the boundary surface of an enclosed region in the calculation of the field exterior to the region. Although the acceptance of this principle has been rather bothersome and not comfortably realized [26

26. S. R. Rengarajan and Y. Rahmat-Samii, “The field equivalence principle illustration of the establishment of the non-intuitive null fields,” IEEE Antennas Propagat. Mag. , **42**, 122–128 (2000). [CrossRef]

*J*∣ in the

_{x}*x*-polarized excitation scheme from the top surface of aluminum oxide,

*I*(AlO

_{SC}_{x}), for the five different interparticle gaps, whereas that from the top surface of Ag nanorod,

*I*(Ag), are shown Fig. 11(b). Notice that the ratio between

_{SC}*I*(AlO

_{SC}_{x}) and

*I*(Ag) at the resonance wavelength increases from about four for

_{SC}*W*= 25 nm to one for

*W*= 5 nm. On the other hand, the area fraction of the Ag nanorod surface increases from 37% for

*W*= 25 nm to 77% for

*W*= 5 nm. The two extracted facts then indicate that the scattering contribution from the ∣

*J*∣ on the top surface of aluminum oxide is consistently larger than that on the top surface of Ag nanorod. The conclusion above is also reached in the

_{x}*y*-polarized excitation scheme, as shown in Figs. 11(c) and (d). Third, the question why the far-field scattering contribution of ∣

*J*∣ is larger than that of

*M*⃗ can be answered by the variation of ∣

*E*⃗∣ and ∣

*H⃗*∣ above the top surface of the Ag nanorod array. As an example, Fig. 12 shows the cross-sectional view of ∣

*E*⃗∣ and ∣

*H*⃗∣ above the top surface of the Ag nanorod array with

*W*= 5 nm. Notice that ∣

*H*⃗∣ extends above the top surfaces of both the aluminum oxide and the Ag nanorod, whereas ∣

*E*⃗∣ concentrates in the gap region and decays in a short distance above the top surface. This behavior indicates that ∣

*E*⃗∣ has a more evanescent-wave character than ∣

*H*⃗∣ does, which is in great contrast to the corresponding near-field behavior of surface plasmon polaritons [30] where both ∣

*E*⃗∣ and ∣

*H*⃗∣ above surface have identical evanescent-wave property. Setälä, Kaivola and Friberg [32

32. T. Setälä, M. Kaivola, and A. T. Friberg, “Evanescent and propagating electromagnetic fields in scattering from point-dipole structures,” J. Opt. Soc. Am. A **18**, 678–688 (2001). [CrossRef]

*E*⃗∣ here. Our calculation, in addition, shows that the surface ∣

*H*⃗∣ in the Ag nanorod arrays makes the dominant contribution in the observed propagating wave in far field, which has been largely ignored in the past. This result may thus impose a need to revise the traditional ∣

*E*⃗∣ -dominated view of the characteristic optical properties of plasmonic structures and raises many interesting questions about the interpretation of the properties of plasmon-based optical devices, such as SERS substrates.

**18**, 491–495 (2006), and references therein. [CrossRef]

33. E. Wolf and J. T. Foley, “Do evanescent waves contribute to the far field?” Opt. Lett. **23**, 16–18 (1998). [CrossRef]

## 4. Conclusions

## Appendix

*ε*

_{exp}and

*ε*

_{DL}are experimental data and the values calculated with the guessed parameters based on Eq. (1), respectively. The

*ω*’s are the discrete frequencies in the spectral range of interest (from 0.2 to 1 μm). The procedure was started by minimizing Ψ according to the genetic algorithm that has a good property of being insensitive to the initial guessed values. As the final values obtained with this algorithm seldom give the best result, nonlinear least square method was then utilized to further minimize Ψ. These two algorithms were then employed iteratively until good results were obtained. In the optimizing process, the least amount of Lorentz terms was used while maintaining minimum error. This approach was followed merely based on the law of parsimony. The extracted values of these parameters are listed as follows:

_{j}*ε*

_{∞}= 3.7,

*ω*= 8.40×10

_{p}^{16}rad/sec,

*τ*= 1.485×10

^{-14}sec,

*f*

_{1}= 0.065,

*ω*

_{1}= 7.79×10

^{15}rad/sec, Γ

_{1}= 3.71×10

^{16}rad/sec,

*f*

_{2}= 0.124,

*ω*

_{2}= 4.25×10

^{16}rad/sec, and Γ

_{2}= 5.73×10

^{15}rad/sec. Figure A1 shows the comparison in dielectric function between the experimental data and the values obtained from the Drude-Lorentz model in the wavelength range from 0.2 to 1.0

*μ*m. Notice that the relative differences in both real and imaginary parts, defined as

*μ*m.

## Acknowledgements

## References and links

1. | S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. |

2. | H. H. Wang, C. Y. Liu, S. B. Wu, N. W. Liu, C. Y. Peng, T. H. Chan, C. F. Hsu, J. K. Wang, and Y. L. Wang, “Highly Raman-enhancing substrates based on silver nanoparticle arrays with tunable sub-10 nm gaps,” Adv. Mater. |

3. | T.-T. Liu, Y.-H. Lin, C.-S. Hung, T.-J. Liu, Y. Chen, Y.-C. Huang, T.-H. Tsai, H.-H. Wang, D.-W. Wang, J.-K. Wang, Y.-L. Wang, and C.-H. Lin, “A high speed detection platform based on surface-enhanced Raman scattering for monitoring antibiotic-induced chemical changes in bacteria cell wall,” PloS One (in press). |

4. | S. Biring, H.-H. Wang, J.-K. Wang, and Y.-L. Wang, “Light scattering from 2D arrays of monodispersed Ag nanoparticles separated by tunable nano-gaps: spectral evolution and analytical analysis of plasmonic coupling,” Opt. Express |

5. | Y. C. Chang, J. Y. Chu, T. J. Wang, M. W. Lin, J. T. Yeh, and J.-K. Wang, “Fourier analysis of surface lasmon waves launched from single nanohole and nanohole arrays: unraveling tip-induced effects,” Opt. Express |

6. | J. Zhao, A. O. Pinchuk, J. M. McMahon, A. Li, L. K. Ausman, A. L. Atkinson, and G. C. Schatz, “Methods for describing the electromagnetic properties of silver and gold Nanoparticles,” Acc. Chem. Res. |

7. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

8. | A. Taflove and S. C. Hagness, |

9. | G. Mie, “Contributions to the optics of turbid media, especially colloidal metal solutions,” Ann. Phys. |

10. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

11. | S. S. Zivanovic, K. S. Yee, and K. K. Mei, “A subgridding method for the time-domain finite-difference method to solve Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. |

12. | H. O. Kreiss and J. Oliger, “Comparison of accurate methods for the integration of hyperbolic equations. Tellus,” |

13. | T. Yamaguchi and T. Hinata, “Optical near-field analysis of spherical metals: application of the FDTD method combined with the ADE method,” Opt. Express |

14. | R. M. Stöckle, Y. D. Suh, V. Deckert, and R. Zenobi, “Nanoscale chemical analysis by tip-enhanced Raman spectroscopy,” Chem. Phys. Lett. |

15. | E. Fort and S. Grésillon, “Surface enhanced fluorescence,” J. Phys. D |

16. | X. Ji, W. Cai, and P. Zhang, “High-order DGTD methods for dispersive Maxwell’s equations and modelling of silver nanowire couping,” Int. J. Numer. Meth. Engng. |

17. | C.-H. Teng, B.-Y. Lin, H.-C. Chang, H.-C. Hsu, C.-N. Lin, and K.-A. Feng, “A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell’s Equations,” J. Sci. Comput. |

18. | D. W. Lynch and W. R. Hunter, “Silver (Ag)” in |

19. | D. W. Thompson, “Optical characterization of porous alumina from vacuum ultraviolet to midinfrared,” J. Appl. Phys. |

20. | J. L. Young and R. O. Nelson, “A summary and systematic analysis of FDTD algorithms for linearly dispersive media,” IEEE Antennas Propag. Mag. |

21. | M. H. Carpenter and C. A. Kennedy, “Fourth order 2N-storage Runge-Kutta scheme,” NASA-TM-109112 (1994). |

22. | M. H. Carpenter, D. Gottlieb, S. Abarbanel, and W. S. Don, “The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: A careful study of the boundary error,” SIAM J. Sci. Comp. |

23. | S. Abarbenel and D. Gottlieb, “On the construction and analysis of absorbing layer in CEM,” Appl. Numer. Math. |

24. | S. Abarbenel, D. Gottlieb, and J. S. Hesthaven, “Well-posed perfectly matched layers for advective acoustics,” J. Comput. Phys. |

25. | M. Born and E. Wolf, |

26. | S. R. Rengarajan and Y. Rahmat-Samii, “The field equivalence principle illustration of the establishment of the non-intuitive null fields,” IEEE Antennas Propagat. Mag. , |

27. | R. S. Elliott, |

28. | M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser & Photon. Rev. |

29. | E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimmers,” J. Chem. Phys. |

30. | L. A. Sweatlock, S. A. Maier, H. A. Atwater, J. J. Penninkhof, and A. Polman, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B |

31. | A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. |

32. | T. Setälä, M. Kaivola, and A. T. Friberg, “Evanescent and propagating electromagnetic fields in scattering from point-dipole structures,” J. Opt. Soc. Am. A |

33. | E. Wolf and J. T. Foley, “Do evanescent waves contribute to the far field?” Opt. Lett. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(290.5850) Scattering : Scattering, particles

(240.6695) Optics at surfaces : Surface-enhanced Raman scattering

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: April 27, 2009

Revised Manuscript: July 7, 2009

Manuscript Accepted: July 13, 2009

Published: July 31, 2009

**Citation**

Bang-Yan Lin, Hui-Chen Hsu, Chun-Hao Teng, Hung-Chun Chang, Juen-Kai Wang, and Yuh-Lin Wang, "Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation," Opt. Express **17**, 14211-14228 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211

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### References

- S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98,011101-011110 (2005). [CrossRef]
- H. H. Wang, C. Y. Liu, S. B. Wu, N. W. Liu, C. Y. Peng, T. H. Chan, C. F. Hsu, J. K. Wang, and Y. L. Wang, "Highly Raman-enhancing substrates based on silver nanoparticle arrays with tunable sub-10 nm gaps," Adv. Mater. 18, 491-495 (2006), and references therein. [CrossRef]
- T.-T. Liu, Y.-H. Lin, C.-S. Hung, T.-J. Liu, Y. Chen, Y.-C. Huang, T.-H. Tsai, H.-H. Wang, D.-W. Wang, J.-K. Wang, Y.-L. Wang, and C.-H. Lin, "A high speed detection platform based on surface-enhanced Raman scattering for monitoring antibiotic-induced chemical changes in bacteria cell wall," PloS One (in press).
- S. Biring, H.-H. Wang, J.-K. Wang, and Y.-L. Wang, "Light scattering from 2D arrays of monodispersed Ag nanoparticles separated by tunable nano-gaps: spectral evolution and analytical analysis of plasmonic coupling," Opt. Express 16,15312-15324 (2008). [CrossRef] [PubMed]
- Y. C. Chang, J. Y. Chu, T. J. Wang, M. W. Lin, J. T. Yeh, and J.-K. Wang, "Fourier analysis of surface plasmon waves launched from single nanohole and nanohole arrays: unraveling tip-induced effects," Opt. Express 16, 740-747 (2008). [CrossRef] [PubMed]
- J. Zhao, A. O. Pinchuk, J. M. McMahon, A. Li, L. K. Ausman, A. L. Atkinson, and G. C. Schatz, "Methods for describing the electromagnetic properties of silver and gold Nanoparticles," Acc. Chem. Res. 41, 1710-1720 (2008). [CrossRef] [PubMed]
- B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994). [CrossRef]
- A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method (Artech House, Boston, 2005).
- G. Mie, "Contributions to the optics of turbid media, especially colloidal metal solutions," Ann. Phys. 25, 377-445, (1908). [CrossRef]
- K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966). [CrossRef]
- S. S. Zivanovic, K. S. Yee, and K. K. Mei, "A subgridding method for the time-domain finite-difference method to solve Maxwell’s equations," IEEE Trans. Microwave Theory Tech. 39, 471-479 (1991). [CrossRef]
- H. O. Kreiss and J. Oliger, "Comparison of accurate methods for the integration of hyperbolic equations.Tellus," 24, 199-215 (1972).
- T. Yamaguchi and T. Hinata, "Optical near-field analysis of spherical metals: application of the FDTD method combined with the ADE method," Opt. Express 15, 11481-11491 (2007). [CrossRef] [PubMed]
- R. M. Stöckle, Y. D. Suh, V. Deckert, and R. Zenobi, "Nanoscale chemical analysis by tip-enhanced Raman spectroscopy," Chem. Phys. Lett. 318, 131-136 (2000). [CrossRef]
- E. Fort and S. Grésillon, "Surface enhanced fluorescence," J. Phys. D 41, 013001:1-31 (2008). [CrossRef]
- X. Ji, W. Cai, and P. Zhang, "High-order DGTD methods for dispersive Maxwell’s equations and modelling of silver nanowire couping," Int. J. Numer. Meth. Engng. 69, 308-325 (2007). [CrossRef]
- C.-H. Teng, B.-Y. Lin, H.-C. Chang, H.-C. Hsu, C.-N. Lin, and K.-A. Feng, "A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell’s Equations," J. Sci. Comput. 36,351-390 (2008). [CrossRef]
- D. W. Lynch and W. R. Hunter, "Silver (Ag)" in Handbook of optical constants of solids, E. D. Palik, ed. (Academic Press, Orlando, 1985), pp. 350-357.
- D. W. Thompson, "Optical characterization of porous alumina from vacuum ultraviolet to midinfrared," J. Appl. Phys. 97, 113511:1-9 (2005). [CrossRef]
- J. L. Young and R. O. Nelson, "A summary and systematic analysis of FDTD algorithms for linearly dispersive media," IEEE Antennas Propag. Mag. 43,61-77 (2001). [CrossRef]
- M. H. Carpenter and C. A. Kennedy, "Fourth order 2N-storage Runge-Kutta scheme," NASA-TM-109112 (1994).
- M. H. Carpenter and D. Gottlieb and S. Abarbanel, and W. S. Don, "The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: A careful study of the boundary error," SIAM J. Sci. Comp. 16,1241-1252 (1995). [CrossRef]
- S. Abarbenel and D. Gottlieb, "On the construction and analysis of absorbing layer in CEM," Appl. Numer. Math. 27,331-340 (1998). [CrossRef]
- S. Abarbenel and D. Gottlieb, and J. S. Hesthaven, "Well-posed perfectly matched layers for advective acoustics," J. Comput. Phys. 154,266-283 (1999). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999), pp. 724-726.
- S. R. Rengarajan and Y. Rahmat-Samii, "The field equivalence principle illustration of the establishment of the non-intuitive null fields," IEEE Antennas Propagat. Mag., 42, 122-128 (2000). [CrossRef]
- R. S. Elliott, Antenna theory and design (Prentice-Hall, New Jersey, 1981), pp. 114-117.
- M. Pelton, J. Aizpurua, and G. Bryant, "Metal-nanoparticle plasmonics," Laser & Photon. Rev. 2, 136-159 (2008). [CrossRef]
- E. Hao and G. C. Schatz, "Electromagnetic fields around silver nanoparticles and dimmers," J. Chem. Phys. 120, 357-366 (2004). [CrossRef] [PubMed]
- L. A. Sweatlock, S. A. Maier, H. A. Atwater, J. J. Penninkhof, and A. Polman, "Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles," Phys. Rev. B 71, 235408:1-7 (2006).
- A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Rep. 408, 131-314 (2005). [CrossRef]
- T. Setälä, M. Kaivola, and A. T. Friberg, "Evanescent and propagating electromagnetic fields in scattering from point-dipole structures," J. Opt. Soc. Am. A 18, 678-688 (2001). [CrossRef]
- E. Wolf and J. T. Foley, "Do evanescent waves contribute to the far field?" Opt. Lett. 23, 16-18 (1998). [CrossRef]

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