## Optical wave properties of nano-particle chains coupled with a metal surface

Optics Express, Vol. 15, Issue 19, pp. 11827-11842 (2007)

http://dx.doi.org/10.1364/OE.15.011827

Acrobat PDF (456 KB)

### Abstract

Optical phenomena supported by ordered and disordered chains of metal nano-particles on a metal surface are investigated by considering a particular example of gold nano-bumps on a gold surface. The TWs supported by these structures are analyzed by studying the frequency-wavenumber spectra of the fields excited by localized sources placed near the chain. Periodic nano-bump chains support traveling waves (TWs) that propagate without radiation loss along, and are confined to the region near, the chain. These TWs are slow waves with respect to both space fields and surface plasmon polaritons supported by the metal surface. For nearly resonant nano-bumps, the TWs are well confined and can be excited efficiently by a localized source placed near the chain but the TW propagation length is short. For non-resonant nano-bumps, the TWs have large propagation lengths but are not well confined and are excited less efficiently. The TWs supported by nano-bump chains were shown to have larger propagation lengths than free-standing chains of the same dimension/size and cross-sectional confinement. TWs also are supported by disordered chains and chains with sharp bends. Perturbations in nano-bump positions are shown to reduce the TW propagation length much less significantly than perturbations in their sizes. Transmission through sharp chain bends is much stronger for nearly resonant nano-bumps than for nonresonant ones. In addition to their ability to support TWs, nano-bump chains can be used to manipulate (excite/reflect/refract) SPPs on the metal surface.

© 2007 Optical Society of America

## 1. Introduction

2. J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider, W. Gotschy, and A. Leitner, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. **82**), 2590–2593 (1999). [CrossRef]

15. C. Girard and R. Quidant, “Near-field optical transmittance of metal particle chain waveguides,” Opt. Express **12**, 6141–6146. [PubMed]

2. J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider, W. Gotschy, and A. Leitner, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. **82**), 2590–2593 (1999). [CrossRef]

15. C. Girard and R. Quidant, “Near-field optical transmittance of metal particle chain waveguides,” Opt. Express **12**, 6141–6146. [PubMed]

2. J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider, W. Gotschy, and A. Leitner, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. **82**), 2590–2593 (1999). [CrossRef]

4. S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss,” Appl. Phys. Lett. **81**, 1714–1716 (2002). [CrossRef]

8. C. R. Simovski and A. J. V. S. A. Tretyakov, “Resonator mode in chains of silver spheres and its possible application,” Phys Rev. E **72**, 066606 (2005). [CrossRef]

10. R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett.41, (2005). [CrossRef]

15. C. Girard and R. Quidant, “Near-field optical transmittance of metal particle chain waveguides,” Opt. Express **12**, 6141–6146. [PubMed]

^{2, 4, 8–10, 16–24}notwithstanding, the study of optical TW guidance and SPP scattering by nano-particle chains residing near metal-dielectric surfaces has received only scant attention.

*on*a metal surface. Ordered and disordered, straight and sharply bent chains of gold parallelepiped shaped nano-particle residing directly on a gold surface, henceforth referred to as nano-bumps, are considered. By investigating frequency-wavenumber spectra of transient fields on chains generated in response to a localized excitation, source-excited as well as source free-fields supported by these structures are characterized. It is shown that nano-bump chains support TWs that are well confined to a region near the chain and have propagation lengths larger than those of TWs supported by free-standing chains comprising identical elements. The TWs are shown to be slow waves with respect to both space and SPP fields, and to be supported efficiently not only by straight ordered nano-bump chains but also by disordered chains and chains with sharp bends.

## 2. Definitions and phenomenology

*x*-axis; the chain’s periodicity is Λ≪

*λ*=

*2πcω*; here

*λ*is the free-space wavelength,

*c*is the free-space speed of light, and

*ω*is the angular frequency. The metal interface coincides with the

*x-y*plane; bumps aside, the space

*z*>0 is air-filled.

*z*>0, the electric field of the lowest order TW can be expressed as

**Ψ**(

*ρ*) describes the field distribution in terms of transverse-to-

_{yz}, φ_{yz}*x*cylindrical coordinates

*ρ*and tan

_{yz}=(y^{2}+z^{2})^{1/2}*φ*=tan

_{yz}^{-1}

*z/y*and

*k*is the (frequency-dependent) TW wavenumber. The field representation (1) indicates that the field propagates along the structure with the wavenumber

_{TW}(ω)*k*and decays out of the chain in the

_{TW}*ρ*dimension with a wavenumber

_{yz}*k*obtained via the relation

_{ρ}*k*+

^{2}_{ρ}*k*

^{2}

_{TW}=(

*ω/c*)

^{2}.

*k*} determines the TW phase velocity as well as the TW loss and confinement near the chain. The TW loss can be associated with dissipation and radiation. The dissipation loss is always present due to the lossy nature of the metals in the optical regime. Two kinds of radiation loss can exist for the considered structure: Radiation loss into space waves out of the interface and radiation loss into SPPs along the interface; the first type of loss also may occur for free-standing chains but the second cannot. For TWs supported by a nano-bump chain to propagate without radiation loss, Re{

_{TW}*k*

_{TW}(

*ω*)} should be greater than both the free-space wavenumber

*k*

_{0}=

*ω/c=2π/λ*and the real part of the (frequency-dependent) SPP wavenumber

*k*(

_{spp}*ω*), i.e.

*k*

_{0}<Re{

*k*} <Re{

_{spp}*k*}; when this condition is met, the TWs are slow waves with respect to both space waves and SPPs and they decay exponentially away from the chain; larger Re{

_{TW}*k*

_{TW}} imply stronger confinement. When Re{

*k*

_{TW}}<

*k*

_{0}, the TWs are fast (or leaky) waves with respect to both space and SPP fields. When

*k*

_{0}=Re{

*k*

_{TW}} < Re{

*k*}, the TWs are fast (or leaky) with respect to the SPPs but slow with respect to space waves. In this paper we focus on the case Re{

_{spp}*k*} < Re{

_{spp}*k*

_{TW}} where no radiation losses occur. An investigation of the fast wave regime will be presented elsewhere.

**82**), 2590–2593 (1999). [CrossRef]

4. S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss,” Appl. Phys. Lett. **81**, 1714–1716 (2002). [CrossRef]

8. C. R. Simovski and A. J. V. S. A. Tretyakov, “Resonator mode in chains of silver spheres and its possible application,” Phys Rev. E **72**, 066606 (2005). [CrossRef]

10. R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett.41, (2005). [CrossRef]

13. A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B **74**, 205436 (2006). [CrossRef]

*ω*(with corresponding resonance wavelengths

_{p}*λ*determined by the nano-bump’s dimensions as well as the nano-bump’s and metal surface’s constitutive parameters. In the mid-infrared regime, where many metals are nearly perfect conductors, bumps that are much taller than wide act as monopoles with resonant wavelengths

_{p}=2πc/ω_{p})*λ≈4*(bump height)/

*q*,

*q*=1, 2,… For frequencies approaching the metal plasma frequency, e.g. in the near-infrared or visible regimes, and depending on the nano-bump shape, resonances can be supported by nano-bumps of deep subwavelength size. Due to the presence of the metal surface, the resonant frequencies of the nano-bumps are substantially lower than the resonant frequencies of free-standing nano-particles of the same shape and size. This, in turn, implies that the TW properties of nano-bump chains are quite different from those of free-standing chains. When a nano-bump is excited by an external field, the bump’s resonance frequencies/wavelengths appear as complex poles in the frequency dependence of the nano-bump’s response to the excitation. Therefore, as the wavelength

*λ*is scanned through

*λ*=Re{

_{r}*λ*}, the magnitude of the field response exhibits strong peaks and its phase exhibits rapid variations.

_{p}^{25}, such condition is obtained for wavelengths greater than the resonant wavelength of the isolated nano-bumps (assuming that the wavelength of operation is near a single resonance wavelength). The TWs supported behave quite differently when the individual nano-bumps are near- or non-resonant. Indeed, resonances lead to strong interactions between the nano-bumps along the chain, which, in turn, leads to large TW wavenumbers and strong TW confinement near the chain [2

**82**), 2590–2593 (1999). [CrossRef]

4. S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss,” Appl. Phys. Lett. **81**, 1714–1716 (2002). [CrossRef]

8. C. R. Simovski and A. J. V. S. A. Tretyakov, “Resonator mode in chains of silver spheres and its possible application,” Phys Rev. E **72**, 066606 (2005). [CrossRef]

10. R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett.41, (2005). [CrossRef]

13. A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B **74**, 205436 (2006). [CrossRef]

## 3. Characterization techniques

*w×w*and height

*h*residing on a gold interface. Gold’s constitutive parameters are described by a Drude model with plasma frequency

*ω*=1.32×10

_{plasma}^{4}THz and damping parameter Γ=1.2×10

^{2}THz [26

26. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science **312**, 892–894 (2006). [CrossRef] [PubMed]

*ω*)

_{0}t*δ*(

**r**-

**r**

_{0}) ; here

**r**

_{0}is the dipole location,

*ω*the carrier frequency, and

_{0}*τ*the pulse’s temporal width; 1/

*τ*determines the pulse’s bandwidth. This dipole models an excitation due to a pin-hole or a near-field probe (e.g. in an NSOM setup).

**E**(

*t,x*) observed (for fixed and sufficiently small

*y*and

*z*) near an

*infinite*nano-bump chain. Assume the field is excited by a broadband dipole placed near the chain. Define the space-time Fourier transform of the field

**E**(

*t,x*) as

**Ê**(

*ω,k*) can be approximated as

_{x}**Ê**(

*ω,k*)=

_{x}**Ψ**(

*ρ*) exp (-(

_{yz},φ_{yz}*k*-(

^{2}_{TW}*ω/c*)

^{2})

^{1/2}

*ρ*) is the TW cross-sectional field distribution and

_{yz}*a*

_{1,2}are the amplitudes of the TWs propagating to the right and left of the source. The TW wavenumbers

_{TW}±

*k*appear as poles in the spectral field

**Ê**(

*ω,k*). It follows that for every

_{x}*ω*, the

*k*-dependence of the spectral field

_{x}**Ê**(

*ω,k*) exhibits peaks. The peak maxima TW (

_{x}*ω,k*≈±Re{

_{x}*k*

_{TW}(

*ω*)}) in the (

*ω,k*) plane are indicative of Re{

_{x}*k*

_{TW}(

*ω*)} ; the peak widths approximate the dependence Im{

*k*

_{TW}(

*ω*)}.

*finite*chains using a finite difference time domain solver 27] that incorporates impedance boundary conditions to account for the presence of the metal surfaces [28

28. L. B. Felsen and N. Marcuvitz, *Radiation and Scattering of Waves* (IEEE Press, Piscataway NJ, 1994). [CrossRef]

*x*=Δ

*y*={-12.5nm-25nm} and Δ

*z*={5nm-10nm} with time discretization of Δ

*t*=0.8Δ

*x/c*. The computational domain was terminated with perfectly matched layers [27].

*D*≫

*λ*that, as before, is excited by a dipole residing near the chain. The chain’s edges affect the TW field distribution along the chain in two ways. First, TWs reflected from the chain edges appear and the left and rightward TWs combine into a standing wave. Second, near the chain ends the field complicated structure and changes rapidly because of edge scattering and the generation of higher-order modes. To eliminate the latter effect, a weighted Fourier transform is introduced via

*w(x)*is a window function that is flat in the center section of the nano-bump chain and gradually approaches zero near its ends. When Im{

*k*}

_{TW}*D*, i.e. when the TW propagation distance is smaller than the chain length, and the source is located far from both chain edges, the chain edges do not have a significant effect and

**Ê**

*(*

_{D}*ω,k*)≈

_{x}**Ê**(

*ω,k*). In this case, both dependencies Re{

_{x}*k*(

_{TW}*ω*)} and Im{

*k*(

_{TW}*ω*)} can be found from the transient field

**E**(

*t,x*) by studying

**Ê**

*(*

_{D}*ω,k*) in (4). When Im{

_{x}*k*}<

_{TW}*D*, i.e. when the TWs propagate without noticeable decay over the length of the chain, then

**Ê**

*(*

_{D}*ω,k*)≈ã

_{x}_{1}ŵ(

*k*(ω))+ã

_{x}-k_{TW}_{2}ŵ(

*kx-k*(ω)), where ã

_{TW}_{1}and ã

_{2}are standing wave amplitudes that characterize both the strength of the excitations of the leftward and rightward propagating TWs as well as their reflections from the two chain edges. In this case, one cannot easily find the dependence Im{

*k*(ω)} from

_{TW}*E(t,x)*and

**Ê**

*(*

_{D}*ω,k*). However, since

_{x}*D*≫

*λ*, the locations of the maxima of

**Ê**

*(*

_{D}*ω,k*) in the (

_{x}*ω,k*) plane still approximate well the dependence TW Re{

_{x}*k*(

*ω*)}. In addition, one can find the standing wave amplitudes ã

_{1}and ã

_{2}.

## 4. Traveling waves on straight ordered nano-bump chains

*w*=100 nm and height

*h*=70 nm, 100 nm, and 150 nm (see insert to Fig. 2); the center of the nano-bumps’ bottom facets coincides with the origin. These nano-bumps were excited by an SPP which, in turn, was excited by a transient dipole with

*ω*

_{0}=1.88×10

^{15}rad/

*s*and

*τ*=5.08×10

^{-14}

*s*residing at

**r**

_{0}=(-8

*µ*m,50 nm,50 nm), i.e., far from the nano-bump. The scattered field was obtained by computing fields in the presence and absence of the nano-bump and subtracting the latter from the former. The spectral content of the scattered field (normalized with respect to the magnitude of the Fourier transform of

**J**(

*t,*)) at

**r****r**=(100 nm,50 nm,100 nm) near the nano-bump is shown in Fig. 2. The scattered field exhibits the resonant behavior as manifested by peak maxima at

*λ*=0.568

*µ*m, 0.755

*µ*m and 1.05

*µ*m for

*h*=70

*nm*, 100

*nm*, and 150

*nm*, respectively. The peak maxima approximate the resonance wavelengths

*λ*. Note that these resonance wavelengths significantly exceed the size of the nano-bumps. As shown below, the wave guiding properties of nano-bump chains significantly depend on the difference between the wavelength of operation and the resonance wavelength of the isolated nano-bumps comprising the chain. Also note that for the chosen structure and excitation parameters the size of the nano-bumps is greater than the skindepth in gold. As a result, the resonance frequencies/wavelengths depend both on the nano-bumps’ shape and size. This is in contrast to the often considered static/quasi-static regime, where the resonant frequencies depend only on the chain elements’ shape [2

_{r}**82**), 2590–2593 (1999). [CrossRef]

13. A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B **74**, 205436 (2006). [CrossRef]

*h*=70nm,

*h*=100 nm, and 150 nm were obtained at

*λ*=0.441

*µ m*,

*λ*=0.476

*µm*, and

*λ*=0.517

*µm*, respectively (for the cross-sectional size of

*w*=100

*nm*).

*N*=160 bumps of width and height

*w=h*=100 nm, and periodicity Λ=200 nm ; the chain length

*D*=32

*µm*(Fig. 1). The chain symmetrically straddles the +x -axis; the center of the bottom facet of the first (leftmost) bump coincides with the origin.

**E**(

*t,x*)| observed in the horizontal plane

*z*=3

*h2*, i.e. 50 nm above the nano-bump’s top face, excited by the above dipole moved to

**r**

_{0}=(16

*µ*m,10

*µ*m,150 nm). The dipole is located sufficiently far from the chain so that the chain, for all practical purposes, is excited by an SPP. It is evident that the chain reflects part of the SPP, while the chain’s TWs are only weakly excited from the chain edges.

**Ψ**(

*ρ*) and wavenumber

_{yz}, φ_{yz}*k*

_{TW}. Figure/snapshot/movie 4 shows the temporal evolution of |

**E**(

*t,x*)| in the

*z*=3

*h*2 plane when the dipole resides as

**r**

_{0}=(-200nm,0,150 nm), that is near the left edge of the chain. Two waves are observed. The wave with a greater velocity corresponds to the SPP supported by the metal surface, whereas the wave with a lower velocity is the TW supported by the chain. The TW exhibits dispersion, i.e. the pulsed field changes spatial width while propagating. In addition, the field has a stronger confinement in the left part of the TW spatial extension, which is due to the fact that slower wave components have a larger propagation wavenumber. The visual reduction of the confinement in the front part of the chain spatial extension is attributed to the reduced excitation of the faster field components. The observed field behavior is characterized in the following results and discussions.

*c*

_{1}at

*x*=16

*µ*m,

*z*=150 nm, and along the chain on the line

*l*

_{1}at

*y*=0,

*z*=150 nm, respectively (Fig. 1). The window function in (4) was chosen as

*w(x)*={erfc[40(

*x*-0.9

*D*)/

*D*]+ erfc[40(0.1

*D-x*)/

*D*]}/4, where erfc(

*x*) is the complementary error function. For

*λ*<0.755 µm, i.e. for wavelengths smaller than the resonance wavelength of the isolated nano-bump

*λ*(of height

_{r}*h*=100 nm), neither propagation nor field confinement is observed. For

*λ*=0.85 µm, i.e., slightly larger than

*λ*, a TW is excited efficiently. The TW field is strongly confined near the chain [Fig. 5(a)] but at the same time it is strongly attenuated while propagating along the chain [Fig. 5(b)]. The strong attenuation is due to the strong dissipation in the metal associated with resonant fields in the chain elements and with strong fields confined to the bumps and metal surface. For

_{r}*λ*=1.1 µm, i.e., for a wavelength significantly larger than the isolated nano-bump’s resonance wavelength, the TW propagates along the chain with a weak attenuation. However, the TW excitation strength is relatively weak and the TW is not well confined near the chain. The decrease of the excitation strength, lack of confinement, and weak attenuation are due to the fact that far from resonance the nano-bumps are weak scatterers. Note that for

*λ*=1.1 µm, i.e., when the TW propagation length is large, the field does not decay monotonously along the chain (Fig. 5(b) but rather has a minimum (at

*x/λ*≈ 7). In addition, there are some ripples in the field dependence. This behavior is obtained because the TW propagation length is significantly larger than the length of the chain. As a result the TW is scattered from the chain ends thus leading to the existence of a standing type TW. Figure 5(c) shows the wavelength dependence of the (normalized) spectral content of the field recorded at three points on the line

*l*

_{1}(Fig. 1). In agreement with Figs. 5(a) and 5(b), no propagation occur for

*λ*<

*λ*. For wavelengths only slightly above

_{r}*λ*, the TW magnitude is very weak. The TW transmission peak is obtained around

_{r}*λ*=0.95

*µ*m. For

*λ*>0.95

*µ*m, the TW magnitude decreases. The decrease of the TW magnitude for

*λ*<

_{r}*λ*<0.95

*µ*m is due to an increase of the loss associated with the proximity of the resonance of the individual nano-bumps. The decrease of the TW field for

*λ*>0.95

*µ*m is due to a decrease of the TW excitation strength associated with weaker TW confinement.

**Ê**

*defined in Eq. (4) as a function of normalized*

_{D}*λ*(or

*ω*) and

*k*. To evaluate the temporal Fourier transform, the time domain response was computed and recorded at the required locations for times long enough so that all resonant fields decayed to a relative value of about 10

_{x}^{-8}. The obtained dispersion curve is located below the dispersion curve of the SPP supported by the metal surface (blue line in Fig. 6) so that the TW is a slow wave with respect to both free-space and SPP fields. In the first Brillouin zone for

*k*Λ<

_{x}*π*, the maximum value of the normalized wavenumber

*k*≈2 for

_{x}/k_{0}*λ*≈0.77

*µm*≈

*λ*For this value of

_{r}*k*, the field is strongly confined near the chain as shown in Fig. 5(a). It was found that

_{x}/k_{0}*k*can be larger for smaller periodicities Λ. In addition, it is evident that the maxima of the dispersion curve are well-pronounced in the range 0.4

_{x}/k_{0}*k*Λ/π

_{x}*<*0.7 and less pronounced otherwise. The decrease of the maxima strength for

*k*Λ/

_{x}*π*<0.4 is due to the decrease of the TW excitation for this range of relatively weak TW confinement near the chain. For

*k*Λ/

_{x}*π*<0.7, the decrease of the magnitude is due to the increased TW decay along the chain caused by resonant dissipation. It should be noted that, unlike the dispersion curves of free-standing chains of nano-spheres of sub-skindepth size [2

**82**), 2590–2593 (1999). [CrossRef]

**81**, 1714–1716 (2002). [CrossRef]

**72**, 066606 (2005). [CrossRef]

## 5. Traveling waves on disordered and bent nano-bump chains

### 5.1 Straight disordered chains

*χ*} with uniform distribution. In the second disordered chain type, the height or the width of the nano-bumps was varied randomly with uniform distribution in the range {

_{L}^{w},χ_{L}^{w}*h*(1-

*χ*),

_{h}*h*(1+

*χ*)} or {

_{L}*w*(1-

*χ*),

_{w}*w*(1+

*χ*)}, respectively. All other parameters were as in Figs. 5 and 6.

_{w}*χ*=

_{L}*χ*0.25. The format of the movies is as in Figs. 4. It is observed that for the chain with randomness in the nano-bump position a TW is still supported efficiently even for the chosen relatively large disorder. On the other hand, the TW decays much faster for the chain with randomness in the height. A behavior similar to that in Fig. 8(b) was obtained in the case of disorder in the nano-bump width; the corresponding results, therefore, are not shown.

_{h}_{1}

*l*for two values of the wavelength

*λ*=0.85

*µ*m (near

*λ*) and

_{r}*λ*=1.1

*µ*m (far from

*λ*) and two values of

_{r}*χ*. The format of the figures is identical to that of Fig. 5(b). It is evident that the propagation length of the TW decreases in all cases [Fig. 9(a)–9(c)] with increase of

_{L}, χ_{w}, χ_{h}*χ*, and

_{L}, χ_{h}*χ*. It is also evident that the disorder has a much stronger effect in the case where

_{w}*λ*is close to

*λ*. It is remarkable, that for the chains with positional disorder [Fig. 9(a)] only for very large

_{r}*χ*does one observes noticeable decrease of the TW propagation length. On the other hand, for the chain with disorder in the height and width [Figs. 9(b) and (c)], even weak disorders lead to significant decrease of the TW propagation length. In addition, TWs on these chains [Figs. 9(b) and (c)] are excited much weaker.

_{L}**Ê**

*as a function of normalized*

_{D}*λ*(or

*ω*) and

*k*for the chains with randomness in the nano-bump position and height, respectively, with

_{x}*χ*=0.25 ; the format of the figures is identical to that of Fig. 6. In Fig. 10(a), the dispersion relation of the chain is well pronounced. The dispersion curve is similar to that of the ordered chain in Fig. 6 with some distortion/widening near

_{L}=χ_{h}*k*Λ=

_{x}*π*and corresponding values of

*λ*(or

*ω*). On the other hand, in Fig. 10(b), the dispersion curve is significantly distorted thus indicating that for chains with disorder in the element size, the TWs are supported much less efficiently. It is also noted that the distortion of the dispersion curve is minor for longer wavelengths where

*k*is small and the TWs are not well confined to the region near the chain. On the other hand the distortion is well pronounced for shorter wavelength where

_{TW}*k*is large and the TWs are well confined to the region near the chain. Chains with disorder in the nano-bump width exhibited a behavior similar to that in Fig. 10(b); the corresponding results, therefore, are not shown.

_{TW}### 5.2 Ordered chains with sharp bends

*N*=75 nano-bumps arranged periodically. The chain parameters are identical to those chosen in Figs. 5–7.

*z*=150 nm. Two types of waves are observed. One is the SPP that propagates along the surface from the exciting dipole as a cylindrical wave and is reflected from the chain. The other wave type is a transient TW supported by the chain. It is evident that the TW has a velocity smaller than the velocity of the SPP and therefore it has a larger time delay. It is further observed that the TW is transmitted through the chain bend rather efficiently. Some field components that are reflected and scattered from the chain end are also present.

*λ*=0.85 µm, i.e. for the wavelength just above the resonance wavelength in Fig. 2, the transmission through the bend is very strong with the ratio of the field before and after the bend of 0.35 (however, as in Fig. 5, the decay rate of the TW is relatively high). The significant transmission is due to the strong confinement within the chain, which leads to primarily near field nature of the interaction between the nano-bumps in the chain. For

*λ*=1.1 µm, i.e. for the wavelength significantly above the resonance wavelength, the transmission through the bend is relatively weak with the ratio between the fields from the two sides of the bend of 0.14 (however, the decay rate is low). The weak transmission is due to reduced confinement of the TW field near the chain. It should be noted that the transmission can be further increased by incorporating special chain elements or modifying the nano-bump arranging near the bend, e.g. making it slightly smoother.

## 6. Summary

## Acknowledgments

## Footnotes

# | These authors contributed equally to this work. |

## References and Links

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2. | J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider, W. Gotschy, and A. Leitner, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. |

3. | M. Quinten, A. Leitner, R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. |

4. | S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss,” Appl. Phys. Lett. |

5. | S. A. Maier, P. E. Barclay, T. J. Johnson, M. D. Friedman, and O. Painter, “Low-loss fiber accessible plasmon waveguide for planar energy guiding and sensing,” Appl. Phys. Lett. |

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

7. | I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, “Dipolar emitters at nanoscale proximity of metal surfaces: Giant enhancement of relaxation in microscopic theory,” Phys. Rev. B |

8. | C. R. Simovski and A. J. V. S. A. Tretyakov, “Resonator mode in chains of silver spheres and its possible application,” Phys Rev. E |

9. | W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B |

10. | R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett.41, (2005). [CrossRef] |

11. | M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B |

12. | V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Phys. Rev. B |

13. | A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B |

14. | F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied in the design of metasurfaces and metamaterials,” Phys. Rev. Lett. |

15. | C. Girard and R. Quidant, “Near-field optical transmittance of metal particle chain waveguides,” Opt. Express |

16. | S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. |

17. | H. Ditlbacher, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Two-dimensional optics with surface plasmon polaritons,” Appl. Phys. Lett. |

18. | Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. |

19. | H. L. Offerhaus, B. E. van den Bergen, M., F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating Focused Plasmons by Noncollinear Phasematching on Functional Gratings,” Nano Lett. |

20. | J. M. Steele, Z. Liu, Y. Wang, and X. Zhang, “Resonant and non-resonant generation and focusing of surface plasmons with circular gratings,” Opt. Express |

21. | H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B |

22. | L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K.M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole array,” Phys. Rev. Lett. |

23. | V. Lomakin, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched in between dielectric slabs,” Phys. Rev. B |

24. | V. Lomakin, N. W. Chen, S. Q. Li, and E. Michielssen, “Enhanced transmission through two-period arrays of sub-wavelength holes,” IEEE Microwave Wirel. Comp. Lett. |

25. | R. E. Collin and F. J. Zucker, |

26. | G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science |

27. | A. Taflove, |

28. | L. B. Felsen and N. Marcuvitz, |

29. | A. Alù and N. Engheta, “On Role of Random Disorders and Imperfections on Performance of Metamaterials,” presented at the 2007 IEEE Antennas and Propagation Society International Symposium, Honolulu, Hawaii, 2007. |

**OCIS Codes**

(160.4670) Materials : Optical materials

(310.6860) Thin films : Thin films, optical properties

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 12, 2007

Revised Manuscript: August 28, 2007

Manuscript Accepted: August 31, 2007

Published: September 4, 2007

**Citation**

Vitaliy Lomakin, Meng Lu, and Eric Michielssen, "Optical wave properties of nano-particle chains coupled with a metal surface," Opt. Express **15**, 11827-11842 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-11827

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### References

- C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (J. Wiley New York, 1983).
- J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider, W. Gotschy, and A. Leitner, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999). [CrossRef]
- M. Quinten, A. Leitner, R. Krenn, and F. R. Aussenegg, "Electromagnetic energy transport via linear chains of silver nanoparticles," Opt. Lett. 23, 1331-1333 (1998). [CrossRef]
- S. A. Maier, P. G. Kik, and H. A. Atwater, "Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss," Appl. Phys. Lett. 81, 1714-1716 (2002). [CrossRef]
- S. A. Maier, P. E. Barclay, T. J. Johnson, M. D. Friedman, and O. Painter, "Low-loss fiber accessible plasmon waveguide for planar energy guiding and sensing," Appl. Phys. Lett. 84, 3990-3992 (2004). [CrossRef]
- S. A. Maier and H. A. Atwater, "Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, 11101-11101 (2005). [CrossRef]
- I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, "Dipolar emitters at nanoscale proximity of metal surfaces: Giant enhancement of relaxation in microscopic theory," Phys. Rev. B 69, 121403 (2004). [CrossRef]
- C. R. Simovski and A. J. V. S. A. Tretyakov, "Resonator mode in chains of silver spheres and its possible application," Phys Rev. E 72, 066606 (2005). [CrossRef]
- W. H. Weber and G. W. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70, 125429 (2004). [CrossRef]
- R. A. Shore and A. D. Yaghjian, "Travelling electromagnetic waves on linear periodic arrays of lossless spheres," Electron. Lett. 41, 578-580 (2005). [CrossRef]
- M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B 62, 16356-16359 (2000). [CrossRef]
- V. A. Markel and A. K. Sarychev, "Propagation of surface plasmons in ordered and disordered chains of metal nanospheres," Phys. Rev. B 75, 085426 (2007). [CrossRef]
- A. Alù and N. Engheta, "Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines," Phys. Rev. B 74, 205436 (2006). [CrossRef]
- F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied in the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004). [CrossRef] [PubMed]
- C. Girard and R. Quidant, "Near-field optical transmittance of metal particle chain waveguides," Opt. Express 12, 6141-6146 (2004). [PubMed]
- S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, "Waveguiding in surface plasmon polariton band gap structures," Phys. Rev. Lett. 86, 3008 (2001). [CrossRef] [PubMed]
- H. Ditlbacher, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, "Two-dimensional optics with surface plasmon polaritons," Appl. Phys. Lett. 81, 1762-1764 (2002). [CrossRef]
- Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing surface plasmons with a plasmonic lens," Nano Lett. 5, 1726 -1729 (2005). [CrossRef] [PubMed]
- H. L. Offerhaus, B. E. van den Bergen, M., F. B. Segerink, J. P. Korterik, and N. F. van Hulst, "Creating Focused Plasmons by Noncollinear Phasematching on Functional Gratings," Nano Lett. 5, 2144-2148 (2005). [CrossRef] [PubMed]
- J. M. Steele, Z. Liu, Y. Wang, and X. Zhang, "Resonant and non-resonant generation and focusing of surface plasmons with circular gratings," Opt. Express 14, 5664-5670 (2006). [CrossRef] [PubMed]
- H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998). [CrossRef]
- L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K.M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole array," Phys. Rev. Lett. 86, 1114-1117 (2001). [CrossRef] [PubMed]
- V. Lomakin, "Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched in between dielectric slabs," Phys. Rev. B 71, 235117 (235111-235110) (2005). [CrossRef]
- V. Lomakin, N. W. Chen, S. Q. Li, and E. Michielssen, "Enhanced transmission through two-period arrays of sub-wavelength holes," IEEE Microwave Wirel. Compon. Lett. 14, 355-357 (2004). [CrossRef]
- R. E. Collin and F. J. Zucker, Antenna theory, Part Two (McGraw-Hill, New York, 1969).
- G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Simultaneous negative phase and group velocity of light in a metamaterial," Science 312, 892-894 (2006). [CrossRef] [PubMed]
- A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, MA, 1995).
- L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway NJ, 1994). [CrossRef]
- A. Alù and N. Engheta, "On Role of Random Disorders and Imperfections on Performance of Metamaterials," presented at the 2007 IEEE Antennas and Propagation Society International Symposium, Honolulu, Hawaii, 2007.

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