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Impact of electric quadrupolar coupling on the optical response of an array of nano-objects
View the table of contents for this issue, or go to the journal homepage for more 2010 J. Phys.: Condens. Matter 22 225301
(http://iopscience.iop.org/0953-8984/22/22/225301)
Yiping Zhao
1Department of Photonics and Institute of Electro-Optical Engineering, Chiao Tung University, Hsinchu 300, Taiwan, Republic of China
2Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA
E-mail:jyhuang@faculty.nctu.edu.tw
Received 23 December 2009, in final form 14 April 2010 Published 20 May 2010
Online atstacks.iop.org/JPhysCM/22/225301
Abstract
Intra- and inter-particle coupling effects are important but have not been properly taken into account in modeling the optical response of an array of nano-objects. In this paper, we present a method to analyze the impact of electric quadrupolar coupling on the optical response of a layer of silver nanorods fabricated with oblique-angle deposition (OAD). Our technique can render the non-locally coupled nano-objects into an array of coarse-grained induced charges. The retrieved polarizability tensor of the silver nanorods exhibits non-zero off-diagonal components, revealing information about the array structure and inter-particle quadrupolar couplings. Thus, a more transparent picture of the correlation between the function and structure of an optical metamaterial can be yielded.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Artificial materials formed by nano-objects with appropriate shapes and morphologies can offer useful optical properties for a variety of applications ranging from sensing [1–7] to photonics [8, 9]. Among the artificial materials, arrays of nanorods with high packing density can yield large optical anisotropy or produce high optical response via couplings to surface plasmon polaritons [8]. By using these artificial materials, a variety of new functionalities including nanometer-scale polarization conversion [8], sub-wavelength imaging [10] and cylindrical superparticles with linearly polarized emission [11] have been demonstrated. To facilitate the design of an appropriate metamaterial, researchers usually relied on the Maxwell–Garnett theory to evaluate the optical response of the optical metamaterial [12,13]. In the past, only the dipolar coupling effect has been taken into account, owing to the dimensions of the nano-objects encountered being much smaller than the optical wavelength used. However, it was found that models including only dipolar coupling may not be
appropriate to describe the optical property of a single metallic nanorod with high aspect ratio [14]. Kullock et al [8] had also pointed out that high-order couplings could be critical in an array of nano-objects.
In this paper, we present a method to analyze the optical response of an array of nano-objects, which can render the non-locally coupled nano-objects into an array of coarse-grained induced charges. Our technique is useful not only for the design of a nanostructure with specific properties but also to yield a more intuitive and transparent picture of the correlation between the function and the structure of the metamaterial used.
Oblique-angle deposition (OAD) belongs to a physical vapor deposition process in which the incident vapor atoms are deposited on a substrate at a large incident angle with respect to the surface normal of the substrate [15]. OAD had been demonstrated to be effective, while inexpensive, for the fabrication of metallic nanorod arrays. More importantly, the size, shape and tilt angle of the nanorods can be easily controlled by adjusting the deposition parameters [16].
J. Phys.: Condens. Matter 22 (2010) 225301 C Y Lin et al
Figure 1.Schematic diagram showing the induced charge distribution on nanorods with an asymmetric distortion by an incident light field (a) along the long principal axis of the nanorods and (b) perpendicular to the nanorods. The blue-coloured arrows and ‘+’ and ‘−’ signs denote the induced dipole moments and charges respectively. The yellow-coloured arrows indicate the directions of magnetic dipoles, which is perpendicular to the current loop. Notice that the magnetic dipoles formed by two neighboring nanorods are antiparallel to each other. Insets describe the relative orientation of the coordinate systems of laboratory and nanorods.
Therefore, OAD has been employed for many new applications in biosensing [1–5]. Multilayer OAD has also been invoked to produce a gradient profile of refractive index for antireflection coating over a broad spectral range [15]. In view of the potential applications, we applied our method to analyze the optical responses of OAD substrates. In our case, the dimensions of the tilted silver nanorods are comparable to the incident wavelength where high-order couplings are not negligible. Our method yielded an effective polarizability tensor with non-zero off-diagonal components, revealing information about the array structure and inter-particle quadrupolar coupling effects.
2. Theoretical analysis of light scattering from a
nanorod array with high packing density
An optical field E incident on an array of nanorods can induce two kinds of electric polarizations with symmetric and asymmetric excitation patterns. An in-phase current distribution produces an electric dipole ( P) that generates the symmetric excitation mode. The asymmetric mode originates from an electric quadrupole (Q↔) and a magnetic dipole ( M ). To depict the optical response of the array, we express the electric displacement vector D as a multipole expansion:
∂t D = ∂tE + 4π(∂t P + c∇ × M − ∂t∇ · Q
↔
+ · · ·). (1) The electrons in a dense array of nano-objects with dimensions comparable to the optical wavelength can experience different optical field strengths, depending on the coordinates of the electrons in the nano-object, array structure and optical
excitation geometry. By utilizing the coarse-grain scheme, we can decompose the distorted charge distribution induced in a nano-object into a set of domains. Inside each domain, the optical field strength can be considered to be constant. Therefore, the induced charge distribution is effectively averaged to form a set of coarse-grained charged particles. The higher-order multipoles can then be viewed as a group of locally coupled coarse-grained charges. The spatial symmetry of the array structure is also automatically embedded into the strengths of the multipoles. This approach not only serves as the foundation for the discrete dipole approximation but also offers an intuitive way to look at equation (1).
Recently, Cho et al [17] had investigated a layer of nanorods with the discrete dipole approximation and discovered higher-order multipoles to be much weaker than the coherent sum of P and Q↔. The dimensions of the nanorods are much smaller than the optical wavelength, thus only one single stack of dipoles was included in their model. Based on their finding, we considered the optical response of our film up to electric quadrupolar couplings. However, because our OAD substrates consist of an array of nanorods with dimensions comparable to the optical wavelength, we had taken into account not only the electric quadrupolar couplings in the array plane but also along the longitudinal axes of the nanorods. The resulting coarse-grain picture of electric quadrupolar couplings is complicated, with a variety of three-dimensional configurations, but can be properly depicted in figures1(a) and (b).
We can express the total electrostatic energy of the system as Wtotal = Wcharge+ Wdipole+ Wq+ · · · with a quadrupolar
Z -polarized light (see figure1(a)) can be expressed as Wq,Z =
(WQ31 + WQ32 + WQ33) = (2 + d1/d2)WQ31, while that
induced by X - (see figure1(b)) and Y -polarized optical field is Wq,X= Wq,Y = (WQ11+ WQ12) = (1 + d3/d2)WQ12.
The magnetic field radiated by a magnetic dipole M can be calculated with [18]
HM= −
ick3eikr
24πr ˆeα× Qα, (3) where Qα = βQαβˆeβ and ˆeα is the unit vector along the α axis of the nanorod’s coordinate system. The direction of
M , which is determined by ˆeα× Qα(ˆeα), is perpendicular to the current loop according to the right-hand rule. The opposite current flows shown in figures1(a) and (b) reduce the magnetic dipole, making M to be negligible. Therefore, we can define an effective polarizability↔αr (ω): Pr(ω) = P−∇·Q ↔ = [↔αP(ω)−ik ↔α Q(ω)∇]· E0= ↔ αr (ω)· E0 (4) to account for the electric multipoles optically induced in the nanorod array. The effective polarizability can be cast into a matrix form of ↔α r= αr 11 αr 12 αr 13 αr 21 αr 22 αr 23 αr 31 αr 32 αr 33 , where αr 11= αp11− i(2αQ11kx∂x+ αQ12ky∂y+ αQ13kz∂z), αr 12 = −iαQ12kx∂y, αr 13= −iαQ13kx∂z, αr 21= −iαQ21ky∂x, αr 22= αp22− i(2αQ22ky∂y+ αQ21kx∂x+ αQ23kz∂z), αr 23= −iαQ23ky∂z, αr 31 = −iαQ31kz∂x, αr 32= −iαQ32kz∂y, αr 33= αp33− i(2αQ33kz∂z+ αQ31kx∂x+ αQ32ky∂y). (5)
Here αr i j denotes a charge distortion along the j direction
induced by an i -polarized incident light. In the simplest case, the polarizability tensor shall have a diagonal matrix form, implying that the electric dipole components of a nanorod can only be induced by an optical field along the same direction of the dipole components. However, due to inter-rod couplings, an electric polarization induced by an optical field component perpendicular to the electric polarization is not negligible,
↔
T=
cosφ sinφ 0
− cos θ sin φ cos θ cos φ − sin θ − sin θ sin φ sin θ cos φ cosθ
. (6) Hereθ and φ denote the angles between the Z and Zaxes and the X and Xaxes, respectively. Therefore, the incident optical field in the coordinate system of the nanorods can be calculated by E= E X EY EZ =T↔· E. (7) The light scattering intensity from the nanorod array thus becomes Iscatt= Ii|ˆes ↔ T−1·↔αr · ↔ T ˆei|2, (8)
where ˆei and ˆes are the unit vectors of the incident and the
scattering fields in the laboratory coordinate system.
To yield further insight into the optical response of the nanorod array, we discuss a connection of the dielectric permittivity (ε) and magnetic permeability (μ) of the layer with the effective polarizability of the coarse-grained particles. As mentioned above, the multiple stack geometry of coarse-grained charges could effectively suppress the axial-vector-like magnetic dipole. This leads to the optical response of the nanorod array to be mainly contributed from the electric dipole and quadrupole. Because of the sub-wavelength-scale inhomogeneity of the medium, optical wave propagation inside the system becomes spatially dispersive and a non-zero local field can only exist with specific k2, which is a solution of the characteristic equation directly from Maxwell’s equations. Based on the formalism developed by Cho et al [17], the electric quadrupole↔αQcan contribute to a magnetic
permeabilityμ−1 = 1 − 4π(ω/c)2 ↔αQ, while the dielectric
permittivity of the film is given by ε = 1 + 4π ↔αP. The
effective polarizabilities↔αPand ↔
αQ of the quasiparticles can
be experimentally determined by light scattering as shown by equation (8).
3. Experimental details of light scattering
measurement
We prepared tilted Ag nanorods on glass substrates by using the oblique-angle vapor deposition method as detailed by Zhao et al [19]. These films consist of a layer of 20 nm Ti and 500 nm Ag films, and an array of tilted silver nanorods with
J. Phys.: Condens. Matter 22 (2010) 225301 C Y Lin et al
Figure 2.(a) SEM image of an OAD Ag nanorod substrate. (b) The azimuthal patterns of the light scattering from tilted Ag nanorods were measured (filled squares). The zero azimuthal angle implies the tilt plane of the silver nanorods to be along the incident plane of the excitation light. The fittings to equation (8) are presented by red solid curves. The p-polarized scattering light was measured with s-polarized incident light. (c) Both of the excitation and scattering fields are p-polarized.
an average length of 900 nm, average diameter of 100 nm and average spacing of 177 nm [19]. Figure2(a) shows an SEM image of a typical OAD film. Those Ag nanorods had an average tilting angle of 73◦with respect to the surface normal of the substrate [20].
For light scattering measurements, a continuous wave (CW) laser with a wavelength of 532 nm was used to excite an OAD substrate. We aligned the polarization of the laser beam to be perpendicular to the incident plane (s-polarized) and then controlled its polarization with a Pockels cell driven at a half-wave voltage of 3.5 kV. This enables the polarization of the incident optical field to be switched easily from s- to p-polarized.
We set the incident angle of the excitation beam at 45◦ to maximize the light scattering intensity according to the finding reported in the literature [21]. We used a microscope objective lens to collect the scattering signal along the substrate normal and analyzed the scattering signal with a polarizer. The resulting optical signal was detected with a photomultiplier tube.
4. Results and discussion
To retrieve the effective polarizability tensor of a silver nanorod in an OAD film, we measured the optical scattering signal from the OAD film as a function of azimuthal angle φ of the substrate. The inclined plane of the nanorod is defined to be the plane formed by the major Z axis of the nanorod and the substrate normal, while the incident plane contains the incident beam and the substrate normal. The azimuthal angleφ is defined to be zero as the inclined plane is aligned with the incident plane. When the polarization vector of the incident or the scattering light is perpendicular to the incident plane, it is indicated as s-polarized, whereas p-polarized is used when the polarization of the incident or the scattering field lies in the incident plane. Figures 2(b) and (c) show the azimuthal patterns of the p-polarized scattering signal from silver nanorods excited by s- and p-polarized light, respectively.
By fitting the two azimuthal patterns to equation (8), the polarizability tensor of the tilted silver nanorods averaged over the illuminated area can be deduced. Without including↔αQ k2
a good fit cannot be obtained. We also need an isotropic term to yield a satisfactory fit, suggesting that in our OAD film the underlying Ag thin film and the Ag nanorod arrays (see figure 1(a)) are coupled together and contribute to the observed optical response. A satisfactory fitting, shown as the solid curves in figures 2(b) and (c), yields the following polarizability tensor: ↔ αr= αr 11 αr 12 αr 13 αr 21 αr 22 αr 23 αr 31 αr 32 αr 33 = V 5.10 −0.24 0 −0.24 5.10 0 5.52 2.42 15.29 . (9) We found thatαr 31 is about twice as large asαr 32, indicating
that our OAD nanorod array deviates significantly from a square lattice that requiresαr 31 = αr 32. The deviation shall
originate from the lack of positional order in the OAD array. As indicated by equation (4), we can decompose↔αr into
↔α Pand ↔α Qas ↔ αP= αP 11 0 0 0 αP 11 0 0 0 αP 33 = V 0.90 0 0 0 0.90 0 0 0 1.15 , (10) ↔α Qk2= k2 αQ11 αQ12 αQ13 αQ12 αQ11 αQ23 αQ31 αQ31 αQ33 = V −2.22 0.240.24 −2.22 00 −5.52 −2.42 −3.10 , (11)
where V is the average volume of coarse-grained quasiparti-cles. As expected, the Xand Ycomponents of↔αPare smaller
than those along the Z axis for a prolate ellipsoidal nano-object [22]. From the ratio of the induced dipolar components, we found that each 900 nm long silver nanorod with a diameter of 100 nm is optically driven to produce about seven coupled dipoles along the longitudinal direction. Therefore, the result 4
αr 31
by an angle of cos−1[αr 31/
α2
r 31+ (αr 11− αr 12− αr 33)2/2].
A similar result was also found with a non-zeroαr 32.
When the dimension of a single silver nanorod is greater than the mean free path of electrons in silver (∼57 nm), the size effect of the dielectric constant is insignificant [23]. Assuming the OAD nanorod array embedded in air, we estimated the resulting dielectric constant to be 1.487 + 0.015i. The real part of the effective dielectric constant of our OAD silver nanorod array is much larger than the imaginary part, indicating that the real part of the dielectric constant dominates the optical response and justifies the real-valued results of equations (10) and (11).
The intra- and inter-rod electric quadrupolar couplings have a contribution to the effective magnetic permeability of the layer. In our nanorod array, there are N quadrupolar units per unit volume with N 1, so the effective magnetic permeability of the film can be deduced to be
μ = 1 N V 0.036 0.004 0 0.004 0.036 0 −0.068 −0.035 0.026 . (12) The non-local coupling effect has a significant influence on the optical response of the nanorod array. At this wavelength, while the diagonal terms of the magnetic permeability remain positive as expected, the off-diagonal components αr 31 and
αr 32become negative.
Recent research on optical cloaking devices or omnidi-rectional retroreflectors used highly ordered array structures to enable new optical functionality [24] or yield a profile of refractive index [25]. The non-local coupling effects in a nanorod array can expand the design of new photonic devices by utilizing the concept of transformation optics [26]. The quadrupolar coupling strength, decreasing with inter-rod distance as r−5 [27], allows us to engineer a proper lattice structure to increaseαr 12. In addition, an increase in the aspect
ratio of nanorods shall lead to increased stacks of induced dipoles. Therefore, by properly tuning the distance between two neighboring nanorods and the length of the nanorod, one shall be able to engineer an appropriate metamaterial structure with a specific optical property.
5. Conclusion
We presented a method to analyze the optical response of a metamaterial formed by a dense array of nano-objects. Our
metamaterial can also be formed by a carefully designed array structure.
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