Coherent control of light propagation via nanoparticle arrays

IOP PUBLISHING

JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 40 (2007) S283–S298

doi:10.1088/0953-4075/40/11/S04

Coherent control of light propagation via nanoparticle arrays Maxim Sukharev and Tamar Seideman Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3113, USA E-mail: [email protected]

Received 9 December 2006, in final form 12 February 2007 Published 16 May 2007 Online at stacks.iop.org/JPhysB/40/S283 Abstract We illustrate the possibility of manipulating light in the nanoscale using the combination of plasmonics physics with concepts and tools developed for coherent control of molecular dynamics. Phase and polarization control are applied to guide electromagnetic energy through metal nanoparticle junctions and control its branching ratios at array intersections. Optimal control theory is applied as a design tool, to develop constructs with desired functionality. We suggest also that nanoplasmonics could be used to make spatially localized light sources with predesigned coherence and polarization properties, which could serve to coherently control individual nano-systems. (Some figures in this article are in colour only in the electronic version)

1. Introduction The interaction of light with noble metal nanoparticles (NPs) and with arrays thereof has inspired scientists for several centuries and continues to offer new and fascinating questions for fundamental research along with new opportunities for applications in nanoscience and nanotechnology. Historically, noble metal particles of sub-diffraction size found applications in the staining of glass windows and ceramic pottery [1]. In modern nanophotonics, such particles are widely researched for applications ranging from novel sensors [2] and medical diagnostics [3] through enhanced solar cells [4] and nonlinear light sources [5] to singlemolecule spectroscopy [6] and 3D atom probe technology [7]. We refer the reader to [8–11] for reviews. In this work we suggest that the field of nanoplasmonics could greatly benefit from ideas and methods developed within the context of coherent control of molecular dynamics, with which the audience of the present special issue is familiar. Developed for control of matter waves, approaches such as two-pathway [12], polarization [13] and optimal [14] control could be applied to the longstanding goal of light guidance and manipulation at the nanoscale. We propose also that nanoplasmonics offers new and exciting opportunities for coherent control. In 0953-4075/07/110283+16$30.00 © 2007 IOP Publishing Ltd Printed in the UK

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particular, it provides potentially an approach to producing nano-scale localized light sources with controllable coherence and polarization properties, which would allow the extension of coherent control to sub-diffraction length scales and its application to single nanodevices or single molecules. In the next section we briefly review the physical principles underlying light interaction with single metal NPs and their ordered arrays. Section 3 discusses the theory and its numerical implementation. We conclude section 3 by testing the accuracy of our numerical approach through direct comparison with experiments. Specifically, we compare measured and computed light interference patters in slit-grove studies, where the experimental accuracy is sufficient to provide a benchmark [15]. The fourth section contains the main message of our work, and includes the application of different coherent control concepts to problems in nanoplasmonics. We begin (section 4.1) by proposing that the polarization and phase properties of the incident light could be applied to the longstanding goal of guiding light in the nanoscale through corners and curvatures. Phase interference is applied in this context also to control the branching ratio of the transported electromagnetic energy at arrays intersections. Next (section 4.2) we introduce optimal control theory as a potential design tool, to make plasmonic nano-constructs with desired properties. Finally, in section 4.3, we apply wave interference to explore new opportunities in plasmonic ‘nanocrystals’—periodic arrays of metallic NPs such as spheres, ellipsoids, nano-wires or their combination. Section 5 concludes this work with an outlook to future theoretical and experimental research. 2. Physical principles The physics underlying the response of metal NPs and their arrays to incident light is familiar from related fields such as surface enhanced Raman spectroscopy [16] and tip-assisted spectroscopies [6]. Similar to metal surfaces and sharp tips, metal NPs enhance incident light at frequencies that depend sensitively on the particle shape and size and on the dielectric constants of both the system and the surrounding medium. The origin of the enhancement is collective coherent excitation of conductive electrons in the particle, leading to buildup of polarization charges on the particle surface. Mathematically, the size, shape and material sensitivity of the plasmon resonance phenomenon is readily appreciated by inspection of Mie’s expression for the extinction cross section of a metallic sphere in the long wavelength limit [11],
3 R 3 εm Im(ε) . (1) σ ∼ λ (Re(ε) + χ εm )2 + Im(ε)2 Here, R is the sphere radius, χ is a polarization factor, which depends on the geometry of the particle (χ = 2 in the case of an ideal sphere), εm is the dielectric constant of the medium surrounding the NP, and Re(ε) and Im(ε) are the real and imaginary parts of the metal NP’s dielectric constant ε, respectively. From equation (1) it is evident that the spectrum is highly sensitive to the geometry of the NP and to the material and embedding medium dielectric constants. The Mie expression (1) predicts also that for materials where the real part of the dielectric constant is negative (as is the case for the nobel metals in the visible part of the spectrum, for instance) and close in absolute magnitude to χ εm , the spectrum will experience resonant enhancement, corresponding to a π/2 phase lag of the collective electronic response with respect to the driving field. The solid curve of figure 1 illustrates the plasmonic response of a silver nanosphere in vacuum, εm = 1. The analogue phenomenon in gold nanospheres is shifted to the red by about 150 nm, due to a difference in the real part of ε(ω). The latter feature, corresponding to strong optical absorption of light in the green part of the

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Figure 1. The shape sensitivity of nanoplasmonics phenomena: plasmon resonance extinction spectra for a sphere (solid), a cube (dash-dotted) and two intermediate structures with increasingly sharp corners (dashed and dotted curves). The sphere limit is readily understood in terms of the Mie theory (section 2), while the sequential development of ‘corners’ leads to a red-shift, line broadening and gradual buildup of structure.

visible spectrum, is the origin of the red colour of the Lycurgus glass cup when viewed in transmitted light. One of the opportunities offered by the wavelength shift of the gold NP resonance with respect to that of a silver NP is discussed in section 4.3. Figure 1 illustrates also the sensitivity of the spectrum to the particle shape and symmetry. By sequentially developing four (slightly rounded, so as to focus on realizable structures) corners, to transform the sphere to a cube, we illustrate the effect of sharp corners and edges on the resonance location and energetic width as well as on its structure. In figure 2 we show the time averaged electromagnetic energy in the vicinity of the structures. Analysis of the nature and origin of the plasmonic modes responsible for the observed spectral features (provided elsewhere [17]) is not necessary for our present purpose. We note only the strong localization of the energy in the vicinity of (smoothed) corners, and the size and shape sensitivity of the plasmonic features. Consider next placing a second NP sufficiently close to the resonantly excited particle. Near-neighbour (mostly dipole) interactions between the particles give rise to coherent oscillations of the free electrons in the neighbour particle, which in this case are induced by the excited dipole, rather than by the light. In an array of NPs, such indirect excitations propagate via near-neighbour couplings, providing a potential mechanism for transmission of EM energy in sub-diffraction length-scales. The realization that noble metal NPs provide spatially- and energetically-localized, strong enhancement of incident light has led to growing experimental [18–22] and theoretical [23–28] interest in the properties and function of metal NPs and their arrays. This interest has resulted in tremendous progress in the fabrication and characterization of noble metal NPs [8, 10, 29–31], and in the numerical prediction of their properties [32–39]. Common fabrication techniques, such as electron beam lithography [29] and the manipulation of randomly deposited NPs using the tip of an atomic force microscope [8], allow the fabrication of various nanoconstructs with nanometre precision and regularity of the particle size and the inter-particle distances [10, 42]. Far- [30] and near- [31] field optical spectroscopies enable direct probing of the spatial characteristics and dispersion relations of plasmon modes in metal NPs. Numerical methods, such as the finite-difference time-domain approach (FDTD) [35] and the discrete

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Figure 2. Two-dimensional xy cuts at z = 0 of the EM energy enhancement distribution, defined as the ratio of time-averaged EM energy calculated in the presence, to that calculated in the absence, of the NP, for the four nanoparticles studied in figure 1 in logarithmic scale. The insets show the structures considered.

dipole approximation [33], have proven capable of reproducing experimental observations [32, 34, 36–38] and were usefully applied to make several interesting predictions [39–41]. Along with the sensitivity of the resonant response to the size, shape and arrangement of the material system, the precise control over these parameters and the ability of numerical methods to make quantitative predictions introduce new opportunities for coherent control, as discussed in the following sections. 3. Theory and numerical implementation The electromagnetic (EM) wave propagation and its interaction with the material system are simulated using a finite-difference time-domain (FDTD) approach [35]. The Maxwell equations read, εeff

∂ E = ∇ × H − J, ∂t

µ0

∂ H  = −∇ × E, ∂t

(2)

where E and H are the electric and magnetic field amplitudes, respectively, J denotes the current density, εeff is the effective dielectric constant (defined below) and µ0 is the magnetic permeability of free space. The optical response of the metallic structures is described within the Drude model, with a frequency-dependent dielectric constant [43]   ωp2 , (3) ε(ω) = ε0 ε∞ − 2 ω + iω

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where ε0 is the electric permittivity of free space, ε0 ε∞ is the infinite frequency limit of the dielectric constant, ωp is the bulk plasma frequency and  is the relaxation rate1 . In the metallic regions of space the material dispersion (3) gives rise to a time-dependent current density that evolves as, ∂ J  = α J + β E, ∂t

(4)

where α = −, and β = ε0 ωp2 . In the surrounding free space εeff = ε0 , α = β = 0 and hence J vanishes. Equation (3) has a simple, physically transparent structure. In particular, it highlights the inherent damping in the optical response of metals (see footnote 1), which results from carrier–carrier scattering and presents a fundamental challenge in nanoplasmonics. Time propagation of the coupled equations (2), (4) is performed by a leapfrogging technique within the auxiliary differential equation method [44]. We employ perfectly matched layers absorbing boundaries [45] of depth ranging from 5 to 16 spatial steps in order to prevent unphysical reflection of outgoing EM waves from the grid boundaries. All calculations are performed in parallel, uniformly distributed among the available processors with respect to the second index (y-dimension) of each component of the EM field and current density, using message passing interface routines. Since the field updates at a mesh point take field values from the surrounding points, exchanges are required at domain boundaries. In the case of two-dimensional structures and TEz polarization (transverse-electric mode with respect to z), for instance, this procedure results in sending and receiving sequences of two vectors, Hz (i, jb − 1) and Ex (i, jb + 1), for each processor, where the index jb denotes a global index that corresponds to the boundary between two consecutive processors (or domain boundaries). In the case of full three-dimensional simulations, the set of equations (2), (4) is partitioned onto N xy-layers, where N is the number of processors. Once optimized with respect to the number of processors and grid points, this parallel strategy leads to efficient memory usage and results in a superlinear speedup regime [41]. For instance, the speedup factor (defined as the ratio of the algorithm execution time on a single node with M processors and the execution time on K processors) reaches 16.4 for M = 16 and K = 256 in the case of a two-dimensional grid of dimension 1536 × 1536, exceeding the ideal speedup limit K/M = 16. An interesting experiment in coherent nanophotonics, which offers sufficient precision to test the accuracy of our numerical approach, is the so-termed slit-groove interference problem [15, 46, 47]. A schematic illustration of the experimental set-up is shown in the inset of figure 3. A horizontally polarized EM plane wave interacts with a thin silver film, in which subwavelength slit and groove separated by a distance d have been etched. The incident light is partially transmitted through the slit, while surface waves are generated by the groove. The total signal detected on the output side of the film consists of both the incident field and the surface waves, exhibiting an interference pattern that depends sensitively on the slit-groove distance. Since the signal relies entirely on wave interference, it provides a stringent test on the accuracy of the calculation. Direct comparison between the measured [15] and simulated interference patterns, presented in the main panel of figure 3, shows gratifying agreement between the observation and our calculation. Extending the simulations to large slit-groove distances, we find that, after a rapid decay of the amplitude within the first three microns from the generating groove, the surface plasmon waves persist with near-constant amplitude over The parameters determining the dielectric constant in equation (3) are ε∞ = 8.926, ωp = 1.7601 × 1016 rad s−1 ,  = 3.0841 × 1014 rad s−1 for silver NPs and ε∞ = 9.84, ωp = 1.3673 × 1016 rad s−1 ,  = 1.0179 × 1014 rad s−1 for gold NPs. 1

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Figure 3. Direct comparison of FDTD simulations with experimental data of surface wave generation by subwavelength slit-groove structures. The inset shows schematically the setup, where the thick dashed contour represents the far field detection contour in the FDTD simulations. The slit and the groove widths are 100 nm, the groove depth is 100 nm, the silver film thickness is 400 nm and the incident wavelength is 852 nm. Interference of waves directly propagating through the slit (red) with waves originating from the groove (blue) and propagating to the slit as surface waves gives rise to the simulated interference pattern shown in the main panel. The solid red circles in the main panel show the experimental intensity as a function of the slit-groove distance [15]. The blue curve shows the FDTD result, where we use the Drude model (see equation (3)) for the frequency-dependent dielectric function.

tens of microns, in accordance with the conventional theory [17] and with recent measurements [15]. 4. Results and discussion 4.1. Phase and polarization control in the nanoscale In this subsection we motivate the application of the phase and polarization properties of laser sources as efficient tools for light transport and manipulation in the nanoscale. To that end we consider a T-junction consisting of silver nanospheres that is excited by a spatially localized source at one end. Inspired by research on coherent control of chemical arrangement channels, we seek to control the branching ratio of EM energy channelled into the two symmetric arms of the junction. A schematic illustration of the set-up envisioned is shown in the inset of figure 4. LS indicates the location of the laser source, and D1 and D2 indicate the location of two detectors. Methods of fabricating junctions of the type envisioned are reviewed in [10] and techniques of exciting the array and of detecting the energy with nanoscale spatial resolution are discussed in [8]. The incident field is of the form   ξ ξ (5) E inc (t) = f (t) ex cos cos(ωt + φ) + ey sin sin ωt , 2 2 where f (t) = E0 sin2 (π t/τ ) is the pulse envelop, τ is the pulse duration, ex(y) is a unit vector along the x(y) axis, ξ and φ determine the field polarization (ξ reduces to the ellipticity for φ = 0), and ω is the optical frequency.

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Figure 4. The time-averaged EM energy collected at two symmetrically placed detectors as a function of the wavelength of the incident light, λinc . The inset shows schematically the T-shaped array of silver NPs, indicating the location of the laser source LS and of two detectors, D1 and D2 . The NPs diameter is 40 nm and the edge-to-edge interparticle distance is 10 nm. The laser pulse duration is 110 fs and the total propagation time is 140 fs. The solid and dashed curves in the main panel correspond to linearly polarized incident field along the x- and y-axes, respectively. The dash-dotted and dotted curves give the energies collected in the upper, D1 , and lower, D2 , detectors, respectively, for left circularly polarized incident field.

The energy domain collective response of the junction is illustrated in the main panel of figure 4, which shows the time averaged EM energy collected at the detectors versus the incident wavelength for different source polarizations. Linearly (x or y) polarized light conserves the symmetry of the junction, hence the equivalence of the observation at the upper (D1 ) and lower (D2 ) detectors for linearly polarized light. By contrast, the definite helicity of circularly (or elliptically) polarized light is capable of inducing the sought asymmetry. The main panel of figure 5 shows (on a logarithmic scale) the ratio of time-averaged EM energies collected in the upper (D1 ) and lower (D2 ) detectors, log10 (W1 /W2 ), as a function of the source parameters, ξ and φ in equation (5). We have that the branching ratio can be varied by four orders of magnitude by phase interference, with the EM energy being guided essentially exclusively into a chosen one of the two geometrically equivalent pathways. The time-resolved dynamics underlying the results of figures 4 and 5 is best illustrated through movies of the EM energy propagation via the construct from the source to the detector2 . The localized source excites plasmon oscillations in the leftmost particle; these oscillations induce a secondary time-dependent EM field, which is in resonance with the plasmons in the next particle, resulting in coherent propagation of the EM energy from one particle to the next along the leg of the junction. Whereas a linearly polarized incident field with its polarization vector along ex , of the type used conventionally in nanoplasmonics studies, generates longitudinal plasmon modes and inevitably suffers heavy losses at the intersection, a properly phased elliptically polarized source generates an optimal superposition of longitudinal and transverse plasmon modes to take the desired turn at the intersection. The insets of figure 5 show two 2

The complete time evolution corresponding to the two scenarios illustrated in the insets of figure 5 can be found at http://www.chem.northwestern.edu/˜seideman/Research/Movie seideman/Research/Movie%201b.htm, respectively.

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(a)

(b)

Figure 5. Logarithm of the branching ratio of the time-averaged EM energy collected in the upper (W1 ) and lower (W2 ) detectors of figure 4, log10 (W1 /W2 ), as a function of the polarization parameters of the incident field, ξ and φ. The incident wavelength corresponds to the blue peak of figure 4. The insets show two snap-shots of the instantaneous EM energy spatial distribution, where the polarization parameters are chosen to channel the energy to the upper (inset (a)) and the lower (inset (b)) arms of the T-structure.

snapshots of the time-resolved response of the junction, corresponding to the values of ξ and φ that maximize (inset (a), ξ = 1.6 rad and φ = 2.7 rad), and minimize (inset (b), ξ = 1.6 rad and φ = 5.8 rad), the W1 /W2 ratio. We find that control rests on an intricate interplay between transverse and longitudinal plasmonic modes, whose relative dominance in  space and time is tuned by the relative phase and magnitude of the E-components, to achieve a desired outcome. Before concluding this section, it is appropriate to remark on two practical concerns. One regards the sensitivity of the control mechanism to small deviations from regularity in the size, shape and arrangement of the particles, and to material dispersion of the medium in which the junction is imbedded. The second regards the extent of losses. To explore the first question, we varied the spheres’ radii in the 20 to 25 nm range. The sensitivity to deviations from regularity was checked by introducing a random size and location deviation of 2 nm for each particle. In addition, we repeated the calculation in a dispersive medium. We found that, whereas the details of the time-dependent dynamics and the shape of the resonance features are sensitive to structural modifications, the control mechanism is robust. Of more general concern is the question of EM energy losses in the course of the light propagation. To explore this problem, we computed the power transmission coefficient, defined as a ratio of the time-averaged EM energy at the leftmost NP to that at either the upper or the lower detection points. For the optimal polarization parameters, the simulated coefficient is found to be 2.5, irrespective of a pulse duration. The coefficient in the case of a linearly polarized source is found to be 5 times larger, as a result of substantial losses at the corner that are circumvented by the field ellipticity.

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4.2. Optimal control as a design tool As pointed out in section 1, the optical properties of the NPs and their arrays depend sensitively on the size, shape and arrangement of particles. Along with the availability of advanced fabrication technologies [10] to produce and arrange metal NPs with high precision and regularity, this sensitivity calls for a systematic approach to the design of plasmonic nanodevices, which goes beyond trial-and-error-based approaches. In this section we propose the application of genetic algorithms to that end. The genetic algorithm (GA) belongs to a class of search techniques inspired by the biological process of evolution by means of natural selection [49]. It is widely applied to construct numerical optimization techniques that perform robustly on problems characterized by ill-behaved search spaces, where standard algorithms, such as gradient-search methods, would typically halt near a local extremum. The stochastic nature of the GA makes it a numerically stable functional optimizer [50]. One of the advantages of the GA is that no numerical derivatives of the fitness function with respect to the model parameters need to be computed. As with most stochastic solutions, however, a large number of evaluations are necessary in order to find a global extremum. If the evaluation is computationally expensive, the forward modelling approach can become impractical. The highly parallel FDTD approach described in section 3 makes the GA application sufficiently efficient to address the problem at hand. For example, a typical execution time of our FDTD codes (the time needed for single evaluation of a fitness function, which is usually defined as the time-averaged EM energy collected in a near-field detector) is less than 1 s on 64 processors. We consider a general problem of interaction of time-dependent EM fields with nanostructures, for which a certain objective depends on a set of adjustable parameters, including both the incident field characteristics (the laser pulse duration, wavelength, polarization, spectral composition, etc) and the material properties (the size and shape of individual NPs, their relative arrangements, etc). The objective itself can be defined in terms of the time-averaged EM energy at some point of space or at a given wavelength, or, more generally, in terms of a pre-defined EM field distribution. As a first test of the proposed approach as a potential tool in nanoplasmonics, we consider a trivial scenario, where the result can be predicted prior to the calculation. To that end we use a linear chain of NPs aligned along the x-axis with a single ellipsoid placed in the middle of the array as shown in the inset of figure 6. The system is excited by a localized light source, LS, of the form of equation (5) with φ = 0. The algorithm seeks to maximize the EM energy on the surface of the right most particle, denoted as D in the inset, and the parameters to be optimized are the source ellipticity, ξ , and the angle, α, between the major axis of the ellipsoid and the x-axis. The source wavelength is fixed at the plasmon resonance of the construct for linear polarization of the source and the ellipsoid aligned along x-axis. The main frame shows the convergence of the GA optimization of the EM energy transfer through the structure to the expected configuration, ξ = 0, α = 0 (to within the accuracy of our calculation). The convergence in this case is rapid, requiring approximately 80 iterations to find the global maximum. Although the exercise is simple and does not generate new knowledge, it illustrates the potential of the approach and tests its efficiency. Due to the symmetry of the problem in the case of a linearly polarized source, ξ = 0, positive and negative values of the angle α are equivalent, which results in degeneracy of the fitness function. It is clear from figure 6, however, that the GA approaches exponentially the global extremum, α = 0. EM energy transfer via linear chains of closely spaced NPs has a wide variety of interesting optical features that make it particularly attractive for phase control, including coherence and

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Figure 6. Two-parameter GA maximization of EM energy transfer through the linear chain of silver NPs illustrated in the inset. The structure is excited by an elliptically polarized, spatiallylocalized laser source, LS, and the time-averaged EM energy is detected at D, on the surface of the rightmost NP. The solid squares in the main panel (referred to the left ordinate) show the evolution of the fitness function (taken to be the time-averaged EM energy at D) with the number of generations. The solid blue circles (referred to the right ordinate) illustrate the corresponding evolution of the angle α between the major axis of the ellipsoid and the horizontal (x)-axis. The wavelength is fixed at the plasmon resonance, λinc = 390 nm, for a linearly polarized source with the ellipsoid aligned along the x-axis. The spheres are of 40 nm diameter, the major and minor axes of the ellipsoid are 35 nm and 10 nm, respectively, the edge-to-edge interparticle distance is 5 nm, the incident pulse duration is 60 fs, the number of individuals per generation is 24, and the number of genes (number of significant digits) is 4.

fs time scale. A major obstacle in practical realization of these constructs as optical devices, however, is large EM losses, due in part to radiative coupling. Transfer of plasmon excitation between neighbour NPs is accompanied by strong EM energy radiation to the far-field. A potential route to decreasing EM losses in future plasmonic nanodevices is the design of optimized hybrid nanostructures that combine nanowires with NPs. We are motivated in part by the realization that surface plasmons on nanowires may propagate for distances of tens of microns with minor losses [18]. An example is shown in figure 7. The inset provides a schematic illustration of the device, where LS denotes the x-polarized, ξ = 0, localized laser source and the detector D is placed on the surface of the rightmost NP. The device consists of seven silver NPs and two silver nanowires. Simulations show that the transmission spectrum exhibits a doublepeak structure, where the amplitudes, resonant wavelengths and widths strongly depend on the nanowire length L and on the distance d between the wires and the chain of NPs. As input parameters for the GA optimizations we use the incident wavelength λinc , the distance d, and the wire length L. Convergence is achieved after about 60 generations, resulting in the optimal parameters λinc = 541.35 nm, d = 84 nm and L = 167.74 nm. We note that d = 40 nm corresponds to contact between the wires and the NPs. The coherent dynamics and the coupling mechanisms underlying this result are unravelled by inspection of the time evolution of the EM energy through the optimized construct. The laser source excites plasmon oscillations in the two leftmost NPs, which bifurcate into two components. The first consists of surface plasmons on both inner and upper faces of the nanowire. The second consists of

Coherent control of light propagation via nanoparticle arrays (a)

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(c)

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Figure 7. Three-parameter GA maximization of the EM energy transported through the hybrid nanostructure depicted in the inset. The optimization parameters are the length of wires, L, the distance d between the wires and the NPs chain, and the incident wavelength, λinc , of the linearlypolarized source (LS). The time-averaged EM energy at D is shown in panel (a) as a function of the number of generations, whereas panels (b), (c) and (d) present the evolution of λinc , d and L, respectively. The numerical and material parameters are the same as in figure 6. The range of parameters used in GA optimizations is the following: λinc ∈ [350 nm; 550 nm ], d ∈ [40 nm; 90 nm], L ∈ [40 nm; 170 nm].

plasmonic excitation propagating along the chain of NPs. The optimal values of d and L, to which the algorithm converges, correspond to the distance and length parameters with which the EM couplings between the NPs and the wires are maximized. Simulations of the power transmission coefficient (the ratio of EM energy at the detector to that at the leftmost particle) show that EM losses in the optimal structure are 10 times smaller than with the wires removed to infinity. To conclude this section, we point out the potential opportunity of designing optimized plasmonics to produce nanoscale light sources with desired coherence and/or polarization properties. We find, for instance, that nonspherical NPs and arrays thereof, excited by a plane wave, modify the polarization of the incident light while focussing it in space. Optimal design of the structure and the mutual arrangement of the particles can thus provide nanosources with prespecified polarization and phase properties. Likewise, a chain of nanospheres of diameters and distances that are individually adjustable can be designed to optimize the plasmonic enhancement and focus it to a specific location in space. We find these possibilities particularly intriguing for the prospect of coherent control of individual nanoscale systems. 4.3. Light trapping and funnelling in plasmonic crystals Similarly interesting are the optical properties and potential applications of plasmonic nanocrystals. By crystals, we refer to ordered one-, two- or three-dimensional arrays of metal

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Figure 8. The transmission coefficient (defined as a ratio of time-averaged EM energy calculated in the presence, to that calculated in the absence of the periodic nanocrystal) as a function of the incident wavelength. The solid curve shows the transmission for a single layer of silver NPs, the dashed curve for two layers, the dotted curve for three layers, the dash-dotted curve for four layers and the dash-dot-dotted curve for five layers. The inset shows the set-up envisioned, where the two dashed vertical lines represent the periodic boundaries and the EM energy is calculated along the horizontal dashed line beneath the structure. The perfect matching boundaries are indicated by PML. The nanocrystal is excited by an x -polarized plane wave pulse of 120 fs duration, emanating from the horizontal line located below the upper PML region.

NPs, periodic in one or more directions, that are collectively excited by an external EM field. It has been demonstrated that metallodielectric photonic crystals show large omnidirectional band gaps along with broadband absorption [51, 52]. Experimental measurements of the optical properties of one- [53] and two-dimensional [54] periodic metal nanostructures have illustrated several interesting features. Periodic arrays of sub-wavelength holes have been also shown experimentally to exhibit surprising optical transmission spectra [55]. In this section we propose a scheme of selective trapping of light in a wavelength-sensitive fashion by periodic and finite plasmonic nanoarrays. Consider first the simple nanocrystal schematically depicted in the inset of figure 8. The structure consists of a periodic array of silver NPs with equal diameters of 40 nm and edge-to-edge interparticle separations of 10 nm. It is excited by an x-polarized EM plane wave propagating along the vertical to the crystal plane, as sketched in the inset of figure 8. We focus on the time-averaged EM energy detected in the far-field zone along the horizontal line shown in dashed in the inset. The transmission through the nanocrystal is expressed in terms of the ratio of the time-averaged EM energy in the presence of the nanoconstruct, Wtotal , and in its absence, Winc . The main panel of figure 8 shows the dependence of the transmission on the incident wavelength, λinc , for five structures with different number of layers. The transmission exhibits a strong minimum at λ  365 nm, whose value decreases exponentially with the number of layers, reaching 10−7 with five layers. Interestingly, the reflection coefficient, calculated in the same manner on the input side of the crystal, exhibits a deep minimum at the same incident wavelength as the transmission. The origin of this effect is the excitation of longitudinal (x-polarized) plasmons in nanoarrays, due to which an incident light, initially propagating along the y-axis, is trapped by the crystal and guided along the horizontal layers perpendicular to the incidence direction.

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Figure 9. As in figure 8 but for a finite bimetallic nanoconstruct, as depicted in the inset. Silver and gold nanospheres are shown as blue and black circles, respectively. The spheres are of 40 nm diameter with an edge-to-edge interparticle distance of 10 nm.

The resonant nature of this phenomenon suggests an opportunity to control light propagation in a wavelength sensitive fashion by adjusting the individual plasmon resonances of each layer of a nanocrystal such that different frequency components are trapped at different spatial locations. Here bimetallic nanocrystals, or hybrid crystals composed of particles of two different shapes are inviting. For instance, the plasmon resonances of silver and gold are separated by almost 150 nm and hence a nanocrystal part of which consists of gold and part of silver NPs will trap a certain wavelength range in one part, and another wavelength range in the other part of the array. A possible design of such a selective plasmonic nanocrystal is depicted in the inset of figure 9. The crystal consists in part of silver (blue circles) and in part of gold (black circles) NPs, which are geometrically identical. The main panel of figure 9 shows the transmission coefficient, Wtotal /Winc , as a function of the incident wavelength, λinc . The transmission exhibits two well-separated absorption bands, centred about λ1  345 nm and λ2  470 nm, corresponding to the plasmon resonances of silver and gold, respectively. At these wavelengths, the incident EM plane wave is decomposed into two spatially localized coherent excitations. This is illustrated in figure 10, where we plot the time-averaged distribution of the ratio, Wtotal /Winc , at the incident wavelength corresponding to the plasmon resonance of silver NPs, λ1 . The incident plane wave, initially symmetric, is broken by the hybrid array into two parts, one of which is guided to the left, while the other propagates to the right. The coherent dynamics and the coupling mechanisms underlying this result are unravelled by inspection of the time evolution of the EM energy through the crystal3 . Whereas the bimetallic nanocrystal exhibits well-isolated resonances, the combination of particles of different shapes (e.g., spheres and ellipsoids) in a periodic nanocrystal offers overlapping resonances, whose properties can be tuned by design of the material shape 3 A movie illustrating EM wave propagation in a bimetallic plasmonic crystal can be found at http://www.chem.northwestern.edu/˜seideman/Research/Movie%201c.htm.

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Figure 10. The time-averaged ratio Wtotal /Winc as a function of x and y for the structure of figure 9 at an incident wavelength λinc = 345 nm, corresponding to the excitation of longitudinal plasmons in the silver NPs.

parameters. We show elsewhere [56] that such crystals provide simple models for study of coupled resonance phenomena and suggest new opportunities for coherent transport and manipulation of light. 5. Conclusions Our goal in the research described in the previous sections has been three-fold. First we introduced coherent control tools as an approach to light transport and manipulation in the nanoscale. In particular, we showed that the polarization and phase properties of the incident laser pulse can serve to guide electromagnetic energy around corners and control its branching ratios at array intersections. We illustrated also the potential of genetic algorithms to tailor the coherence properties of the material arrays, serving as a design tool for plasmonic nanoconstructs with desired functionality. Next we inquired into the information content of the structures determined through the control algorithm regarding the physics of light propagation via NP arrays. Finally, and possibly most interestingly, we suggested the possibility of using plasmonics to make spatially-localized light sources with controlled properties. While the challenge is evident, the potential of applying coherent control tools to individual molecules or nanodevices well below the diffraction limit is intriguing. It is appropriate to close this work by remarking that the experimental machinery required to combine coherent control with nanoplasmonics is currently being developed in several laboratories [57]. Acknowledgments We thank the National Energy Research Scientific Computing Center, supported by the Office of Science of the US Department of Energy under contract no. DE-AC03-76SF00098, and San Diego Supercomputer Center for computational resources. References [1] ‘Stained glass’ Encyclopaedia Britannica Online 2006 http://www.britannica.com/eb/article-9108357/stainedglass [2] Alivisatos P 2004 Nature Biotechnol. 22 47

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