NANO LETTERS
Phase and Polarization Control as a Route to Plasmonic Nanodevices
2006 Vol. 6, No. 4 715-719
Maxim Sukharev and Tamar Seideman* Departments of Chemistry and Physics, Northwestern UniVersity, 2145 Sheridan Road, EVanston, Illinois 60208-3113 Received December 17, 2005; Revised Manuscript Received February 7, 2006
ABSTRACT We extend the concepts of phase, polarization, and feedback control of matter to develop a general approach for guiding light in the nanoscale via nanoparticle arrays. The phase and polarization of the excitation source are first introduced as tools for control over the pathway of light at array intersections. Genetic algorithms are next applied as a systematic design tool, wherein both the excitation field parameters and the structural parameters of the nanoparticle array are optimized to make devices with desired functionality. Implications to research fields such as single molecule spectroscopy, spatially confined chemistry, optical logic, and nanoscale sensing are envisioned.
Photonic crystals and planar waveguide technology have long enabled the guidance of electromagnetic (EM) energy at optical frequencies in the microscale. The development of an analogue that would apply to the nanodomain is nontrivial, since it requires one to bypass the inherent limits defined by the diffraction of light. It is nevertheless a rewarding challenge, due to both the new fundamental questions associated with light propagation in subdiffraction scales and the variety of potential applications of optical nanodevices.1,2 These include novel approaches to nanolithography,3 length calibration,4 nanoscale sensing,5,6 spectroscopy,7 and medical diagnostics,8 to mention but a few. A method to induce light propagation in the nanoscale was proposed already several years ago,9 based on the near-field dipole interactions between metal nanoparticles (NPs) arranged in an array. The underlying physics is general and well understood. Noble metal NPs enhance light in a shape- and size-sensitive fashion, due to poles in the particle ac polarizability at its (shape- and size-dependent) plasmon resonance frequency.1,2 In a metal NP array, the surface plasmon resonance enhances the interaction between adjacent particles and may thus give rise to collective response to external light, where near-field coupling between closely spaced particles sets up coupled plasmon modes.10 Tremendous progress has been made on the problem of light interaction with metal NP arrays with the development of methods of fabricating and characterizing such arrays,11 approaches of exciting and probing them locally in space with nanometer resolution,12 and numerical algorithms of predicting their optical properties.13,14 Particularly intriguing are recently developed techniques that provide, along with * Corresponding author. chem.northwestern.edu. 10.1021/nl0524896 CCC: $33.50 Published on Web 02/25/2006
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subwavelength spatial resolution, also subfemtosecond temporal precision, thus extracting dynamical information about collective and single particle plasmon excitation.15,16 Several challenges, however, have for some time remained outstanding.1 One question is how to choose the size, shape, and relative arrangement of the NPs so as to guide the EM energy to a specific location. Another question is how to design the array so as to minimize losses. Even the elementary (but nontrivial) problem of inducing light to bend around corners was found difficult, if possible,17 despite effort. Recent reviews point also to the need for new approaches of understanding the underlying fundamentals.2 It has been recognized that in order to further advance the field a new concept is necessary. Here the tools of phase and polarization control of molecular physics18-20 present an opportunity. Coherent control approaches have been applied in recent years to problems ranging from atomic physics and gas-phase molecular dynamics through solid-state physics and semiconductor device technology to solution chemistry and biology.18-20 While use of the intensity and coherence properties of lasers for control is widespread, recent research on feedback control of molecular dynamics has utilized also their polarization property.21 In the related field of light enhancement by a metal tip (applied, e.g., in single molecule spectroscopy experiments),22 the role of the polarization was recently demonstrated.23 Means of extending coherent control methods to nanoplasmonics are conceptually interesting. On one hand, plasmonic nanodevices open a new class of problems with associated new challenges for coherent control. On the other, plasmon-based enhancement introduces an opportunity to address with light single molecules and single devices and hence a potential to extend coherent optical control from an ensemble method to the domain of
individual nanoscale systems. The practical interest in introducing coherent control as a tool for nanoplasmonics is clear from the review articles, refs 1,2 and the recent advances exemplified, e.g., in refs 3-8, which detail a range of imaginative applications of nanoplasmonics and indirectly suggest others. In the present work we illustrate the possibility of using the phase and polarization properties of light to make new forms of plasmonic nanodevices and to better understand the material properties that underlie the transport of electromagnetic (EM) energy via metal NP arrays. First we introduce polarization control as a general method of producing superpositions of transverse and longitudinal plasmonic modes whose (controllable) phase relation allows guiding of light through multiple branching intersections. Next we illustrate that genetic algorithms can be applied as a systematic design tool, to optimize not only the field parameters but also the material size, shape, and arrangement so as to make devices with desired functionality. Finally we show the possibility of using coherent control techniques to better understand the optical properties of interacting NPs. The energy transport is simulated using a finite-difference time-domain propagation scheme14,25 to solve the Maxwell equations eff
∂E B ) rotH B-B J ∂t
µ0
∂H B ) -rotE B ∂t
∂J B ) RJ B + βE B ∂t
(1)
where B E and H B are the electric and magnetic fields, respectively, and B J denotes the current density. In the metallic regions, the parameters eff ) 0(ω f ∞), R ) -ΓD, and β ) 0ωD2 describe the metallic frequency-dependent dielectric permittivity, , within the Drude model.26 In the surrounding vacuum, R ) β ) 0 and the current vanishes. Here (ω f ∞) is the dielectric constant in the infinite frequency limit, ΓD accounts for electron relaxation processes, and ωD is the bulk plasmon frequency. We use the cylindrical symmetry of the experiment in mind to restrict attention to two (x and y) B E components and one (z) H B component (TEz mode). As an incident source we implement a pointwise elliptically polarized electric field ξ ξ e x cos cos(ωt + φ) + b e y sin sin(ωt) B Einc ) E(t) b 2 2
[
]
(2)
where E(t) ) E0 sin2(πt/τ) is the pulse envelop, τ is the pulse duration, b ex(y) is a unit vector along the x(y) axis, ξ and φ specify the polarization vector (ξ reducing to the ellipticity for φ ) 0), and ω is the optical frequency. Figure 1 illustrates the possibility of guiding EM energy into one or the other of the two branches of a symmetric T-structure solely through wave interference. The inset depicts schematically the NP array, indicating the positions 716
Figure 1. Polarization and phase control. Logarithm of the ratio W1/W2 of the EM energies in the upper (D1) and lower (D2) ends of a T-structure (see inset) as a function of the laser parameters (eq 2). See refs 27 and 30 for the system, field and computational parameters. The pulse is centered at a plasmonic resonance wavelength, λ = 300 nm. Note that D1 and D2 are located at the centers of upper and lower nanoparticles, respectively.
of the light source (LS) and the two detectors (D1, D2). The field, computational, and structural parameters required to reproduce our results are collected in ref 27. Different methods of fabricating such structures are reviewed in ref 11, and means of exciting and probing them with nanoscale spatial resolution are described, e.g., in ref 29. The main frame of Figure 1 shows (on a logarithmic scale) the ratio of the time-averaged EM energy collected at the upper and lower detection points, ln(W1/W2), as a function of the polarization parameters, φ and ξ in eq 2. We have that the branching ratio can be varied by 2 orders of magnitude through phase interference, with the EM energy being guided essentially exclusively into either branch (whereas with pure x- or y-polarization the branching ratio in such symmetric arrays is clearly W1/W2 ) 1). Our results suggest a potential nanoscale plasmon switch or inverter. The results of Figure 1 and similar simulations are robust toward small variations in the frequency as well as toward small deviations of the particle size and spacing from regularity.30 (In the context of Figure 1, the ellipsoids are nonessential and could be replaced by spheres without altering the essentials, although in other contexts the ellipsoidal structure plays an important role.) The physics underlying the results of Figure 1 and related arrays is unraveled by study of the time evolution of the EM energy via the device. The perfect control in the subdiffraction scale exhibited in Figure 1 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 B E components, to achieve a desired outcome. Thus, whereas linearly polarized light in the x direction (Figure 1, inset) excites longitudinal modes, and linearly polarized light in the y direction excites transverse modes, the elliptical polarization excites a superposition of both modes whose detailed composition is determined by ξ and φ. In the example of Nano Lett., Vol. 6, No. 4, 2006
Figure 2. Optimal control as a design tool of a linear chain. The inset in each of the two panels shows schematically the array studied, indicating the location of the source (LS) and the detector (D). The main frame provides the time-averaged EM energy at D, illustrating the convergence properties of the genetic algorithm. (a) Two parameter maximization of the time-averaged EM energy at the surface of the end particle (D), optimizing ξ and the angle R between the ellipse major axis and the x axis. The wavelength of the incident field corresponds to a plasmon peak for the structure depicted in the inset. (b) Three parameter maximization of the timeaveraged EM energy at the center of the end particle (D), optimizing ξ and the two angles R1 and R2 between the major axes of the two ellipses and the x axis. The wavelength (λ ) 350 nm) is chosen arbitrarily.
Figure 1, the ellipticity allows efficient bending about the corner while the sign of φ determines the sense (up vs down) of the energy transport. The manner through which coherent control breaks the symmetry of the junction to induce unidirectionality is thus qualitatively different from that underlying related symmetry breaking phenomena as, e.g., illustrated in ref 31. Polarization control of plasmonic devices is a general effect that we found also in other constructs and which was not recognized before. Well-known, by contrast, is the sensitivity of the plasmon resonance frequency to the size, shape, and arrangement of the NPs. Along with the availability of advanced technology to produce and arrange metal NPs with high precision and regularity (controllability of the radius of a single nanoparticle to within better than 1 nm has been communicated),1 this sensitivity calls for a systematic approach to the design of nanoplasmonic devices that would go beyond trial-anderror-based approaches. Figures 2 and 3 propose the application of genetic algorithms to that end. Panel a of Figure 2 considers a trivial case, where the output parameters could have been predicted, and is included to explore the ability of optimal control to make a design tool and to test its convergence properties. The algorithm seeks to maximize the time averaged EM energy at the detector (D) and the parameters to be optimized to that end are the ellipticity of a laser source of the form (2) (we choose φ ) 0 in this example, with which the ellipticity is given by ξ) and the angle R between the major axis of the ellipsoid and the x axis (see inset). As demonstrated in the main panel, the Nano Lett., Vol. 6, No. 4, 2006
Figure 3. Optimal control as a design tool of a hybrid construct. Two parameter maximization of the time-averaged EM energy at D, optimizing the angle ξ and the distance d between the rod and the chain of circles (see the inset). The upper curve shows the fitness function, F, and the lower gives the distance d vs the number of generations. The wavelength (λ ) 373 nm) corresponds to a plasmon resonance for a range of d values.
algorithm converges within 20 iterations, finding a global maximum at (ξ ) 0, R ) 0). Figure 2a considers a central frequency that corresponds to one of the plasmon resonances of the device. More challenging (and more relevant in practice, since the resonance frequency depends on the structural parameters) is the case of off-resonance excitation, examined in Figure 2b. Here we seek to maximize the EM energy transfer through the construct using as parameters the laser ellipticity and the angles R1 and R2 between the major axes of the two ellipsoids and the x axis. The algorithm converges within 70 iterations, giving an optimal configuration of R1 ) 1.7 and R2 ) 40°, with an ellipticity of 4°. In this example the algorithm converges to the specific configuration for which a plasmon resonance is located at the arbitrarily chosen wavelength. It suggests a means of funneling the response to a preselected wavelength, with potential application, e.g., for nanoscale sensing. The studies leading to parts a and b of Figure 2 suggest also that an optimal control calculation that varies the wavelength along with a structural parameter would locate the optimal structure required to achieve a desired function irrespective of the frequency, i.e., would find an optimal resonance over the structural parameter space. Of the broad variety of other arrays that could be optimized in an analogous fashion, we mention the self-similar chain of ref 32. Figure 3 explores the possibility of alleviating the adverse effect of losses by optimally designing hybrid constructs, which combine NPs with nanorods. One of our motivations is the interest in coherent propagation via multiple interfering pathways. Another motivation is recent experimental studies that illustrate that rods and films are considerably less vulnerable to losses than NP arrays.33-35 The energy spectrum of the array depicted schematically in the inset exhibits two 717
resonances, centered ca. 100 nm apart. Fixing the wavelength at one of the resonances and choosing as variables for the genetic algorithm the source ellipticity and the distance between the rod and the particle chain, d, we obtain the fitness function F shown vs the number of generations in the top panel of Figure 2c. The rod-chain distance, provided in the lower panel, exhibits a nontrivial dependence on the number of generations, converging, after 25 generations, to an optimal distance of d ) 56.6 nm. (We note that 50 nm corresponds to contact between the rods and circles.) No less interesting is the finding that pure x polarization best carries the excitation (at the wavelength considered), the calculation converging to zero ellipticity within 10 iterations. Inspection of the time evolution of the EM energy via the array corresponding to the optimal structure unravels the underlying mechanism. The laser source excites coherent collective oscillations of free electrons in the first two metal NPs, which bifurcate to excite plasmons on the surface of the rods. These excitations travel along the rods and finally excite plasmon oscillations in the last two NPs of the structure. The optimal distance d obtained by the algorithm corresponds to the distance with which the EM coupling between the nanospheres and nanorods is maximized. By contrast, for large values of d, the EM energy propagates mostly via the particle chain. In independent calculations we find the d dependence of the power transmission coefficient (the ratio of EM energy at the detector to that at the left-most particle) to have a maximum at the optimal distance, being a factor of ca. 2.1 lower at the maximum than in the absence of the rods. Figure 3 suggests a potentially general approach to minimizing losses by specifically tailoring nanorod-nanoparticle interactions in hybrid arrays. It is relevant to remark that our scheme does not require ultrashort pulses or pulse chirping, as it relies neither on maintenance of coherence of the electronic response nor on a broad bandwidth. Since it is based on resonance interactions, the control improves as the pulse duration increases, saturating when τ reaches the natural lifetime. A conceptually different, and largely complementary, coherent control approach to the related problem of nanoscale localization is presented in refs 36 and 37, based on pulse chirping. The discussion of Figures 1-3 suggests, and the analysis of a variety of related configurations confirms, that coherent control approaches can provide a wealth of new information about the mechanism of EM energy propagation via NP arrays. For instance, inspection of the time evolution of EM energy via the optimal array furnished by the calculation of Figure 1 provides the superposition of longitudinal and transverse plasmon modes produced by the properly phased components of the elliptically polarized field that guides the light to take the desired turn at the intersection. The outcome of Figures 2 and 3 reveals information about the balance of near- and far-field interactions between nanoparticles and about loss mechanisms. The value of such experiments (both numerical experiments, where the system and field parameters are tuned, and laboratory experiments, where only the field parameters are tuned) may thus go beyond the practical potential to make new and improved plasmonic nanodevices. 718
In summary, we extended concepts of coherent control to guide EM energy in the nanoscale and to derive new information regarding transport mechanisms. Our results suggest the potential role of the polarization and phase properties of the excitation source in future plasmonic nanodevices, such as switches and logic elements. The range of potential applications is clearly vast. One inviting opportunity is the use of coherent control to optimize the light enhancement in tip-enhanced spectroscopy experiments,38 where properly shaped and arranged metal NPs in conjunction with the metal tip could greatly improve the sensitivity. Likewise intriguing is the possibility of controlling or triggering chemical reactions of molecules adsorbed onto metal NPs through spatially controlled field enhancement. The ability of coherent control to localize EM energy in space could find applications in the design of nanoscale sensors.39 The strongly spatially inhomogeneous field produced through plasmonic enhancement could be used to focus and guide physisorbed molecules in the nanoscale. Likewise interesting is the possibility of using this approach to design laserassisted three-dimensional (3D) atom probe experiments, an emerging variant of the 3D atom probe, where a fine needle is combined with a laser pulse to provide atomic-scale imaging of materials.40 The extensions of our approach to nonlinear phenomena, to materials other than noble metals, to nanohole arrays, and to composite NP constructs are inviting. Acknowledgment. We thank Dr. S. Gray, Dr. G. Cheng, Dr. A. Pinchuk, Professor R. P. Van Duyne, and Professor G. Schatz for interesting conversations and the National Energy Research Scientific Computing Center, supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098, and San Diego Supercomputer Center, under Grant No. PHY050006N, for computational resources. References (1) For several of the many available reviews see: (a) Kelly, M.; et al. J. Phys. Chem. 2003, 107, 668. (b) Link, S.; El-Sayed, M. A. Annu. ReV. Phys. Chem. 2003, 54, 331. (c) Barnes, W. L.; et al. Nature 2003, 424, 824. (d) Van Duyne, R. P. Science 2004, 306, 985. (2) For a review see: Hutter, E.; Fendler, J. H. AdV. Mater. 2004, 16, 1685. (3) Liu, Z.-W.; et al. Nano Lett. 2005, 5, 957. (4) Reinhard, B. M.; et al. Nano Lett. 2005, 5, 2246. (5) Rindzevicius, T.; et al. Nano Lett. 2005, 5, 2335. (6) Lu, Y.; et al. Nano Lett. 2005, 5, 119. (7) Genov, D. A.; et al. Nano Lett. 2004, 4, 153. (8) El-Sayed, I. H.; et al. Nano Lett. 2005, 5, 829. (9) Quinten, M.; et al. Opt. Lett. 1998, 23, 1331. (10) Maier, S. A.; et al. AdV. Mater. 2001, 13, 1501. Maier, S. A.; Atwater, H. A. J. Appl. Phys. 2005, 98, 011101. Stockman, M. I.; et al. Phys. ReV. Lett. 2002, 88, 067402. (11) Craighead, H. G.; Niklasson, G. A. Appl. Phys. Lett. 1984, 44, 1134. Hoogenboom, J. P.; et al. Appl. Phys. Lett. 2002, 80, 4828. McMillan, R. A.; et al. Nat. Mater. 2002, 1, 247; Mu¨ller, T.; et al. Mater. Sci. Eng., C 2002, 19, 209. (12) Maier, S. A.; et al. Proc. SPIE 2001, 4456. Quidant, R.; et al. Phys. ReV. B 2004, 69, 085407. (13) Ziolkowski, R. W.; Tanaka, M. J. Opt. Soc. Am. A 1999, 16, 930. (14) Gray, S. K.; Kupka, T. Phys. ReV. B 2003, 68, 045415. (15) Kubo, A.; Onda, K.; Petek, H.; Sun, Z.; Jung, Y. S.; Kim, H. K. Nano Lett. 2005, 5, 1123. (16) Stockman, M. I.; Hewageegana, P. Nano Lett. 2005, 5, 2325.
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(17) Brongersma, M. L.; et al. Phys. ReV. B 2000, 62, R16356. (18) Rabitz, H. A.; et al. Science 2004, 303, 1998. Rabitz, H. A.; et al. Science 2000, 288, 824. Shi, S. H.; Woody, A.; Rabitz, H. J. Chem. Phys. 1988, 88, 6870. (19) Shapiro, M.; Brumer, P. Principles of the quantum control of molecular processes; Wiley-Interscience: Hoboken, NJ, 2003. Brumer, P.; Shapiro, M. Annu. ReV. Phys. Chem. 1992, 43, 257. Shapiro, M.; Brumer, P. J. Chem. Phys. 1986, 84, 4103. (20) Rice, S. A.; Zhao, M. Optical Control of Molecular Dynamics; John Wiley & Sons: New York, 2000. Rice, S. A.; Gordon R. J. Annu. ReV. Phys. Chem. 2000, 48, 601. Tannor, D. J.; Kosloff, R.; Rice, S. A. J. Chem. Phys. 1986, 85, 5805. (21) Brixner, T.; Gerber, G. Opt. Lett. 2001, 26, 557. Brixner, T.; Gerber, G. Chem. Phys. Chem. 2003, 4, 418. (22) Sanchez, E. J.; Novotny, L.; Xie, X. S. Phys. ReV. Lett. 1999, 82, 4014. (23) Brixner, T.; et al. Phys. ReV. Lett. 2005, 95, 093901. (24) Imura, K.; et al. Chem. Phys. 2005, 122, 154701. (25) Taflove, A.; Hagness, S. C. Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed.; Artech House: Boston, MA, 2000. (26) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles, 2nd ed.; Wiley: New York, 1983. (27) In all simulations we consider silver nanoparticles surrounded by vacuum. The radius of the spheres is 25 nm, the semiaxes of the ellipsoids are 35 and 25 nm, and the center-to-center distance between spherical particles is 75 nm. In the wavelength range considered (∼300-370 nm), the dielectric constant of silver is described by the following set of parameters: (ω f ∞) ) 8.926, ωD ) 11.585 eV, ΓD ) 0.203 eV.28 We use two-dimensional grids ranging from x, y ) -850 to 850 nm, with an exponential damping band of 300 nm applied at each edge. The results are converged to within 0.5% with spatial grid steps dx ) dy ) 1 nm and a temporal step dt ) dx/x3c, where c denotes the speed of light in a vacuum. In the genetic algorithm calculations, the number of individuals per generation for
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(28) (29) (30)
(31) (32) (33) (34) (35) (36) (37) (38) (39) (40)
each parameter is 20 and the number of genes (number of significant digits in each individual) is 6. The EM energy detected depends on whether the signal is collected in the center of the end particle or on its surface (and is larger on the surface). Both configurations are considered here since previous calculations used the former, whereas experimental detection methods are consistent with the latter. A pulse duration of 15 fs is used in the calculations shown. The results are weakly τ-dependent, with control improving with τ and saturating as τ converges to the resonance lifetime. (Our use of short pulses is for numerical reasons, as the cost of the calculation increases with the propagation time.) Lynch, D. W.; Hunter, W. R. In Handbook of Optical Constants of Solids; Palik, E. D. Ed.; Academic: Orlando, FL, 1985; pp 350357. Wiederrecht, G. P. Eur. Phys. J. Appl. Phys. 2004, 28, 3. The sensitivity of the results to the particle size was checked by varying the spheres radii in the 25-30 nm range. The sensitivity to deviations from regularity was checked by introducing a random size and location deviation of 2 nm for each particle. The results were found robust. See, e.g.: Bhat, R. D. R.; Sipe, J. E. Phys. ReV. B 2005, 72, 075205 and references therein to earlier work on the same topic. Li, K.; et al. Phys. ReV. B 2005, 71, 115409. Dickson, R. M.; Lyon, L. A. J. Phys. Chem. B 2000, 104, 6095. Wang, K.; Mittleman, D. M. J. Opt. Soc. Am. B 2001, 22, 2001. Rokitski, R.; et al. Phys. ReV. Lett. 2005, 95, 177401. Li, K.; Stockman, M. I.; Bergman, D. J. Phys. ReV. Lett. 2003, 91, 22740. Lee, T.-W.; Gray, S. K. Phys. ReV. B 2005, 71, 035423. Hla, S.-W. J. Vac. Sci. Technol., B 2005, 23, 1351. Haes, A. J.; Van Duyne, R. P. Expert ReV. Mol. Diagn. 2004, 4, 527. Deconihout, B.; et al. Surf. Interface Anal., in press.
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