Linear and nonlinear optics of hybrid plasmon-exciton nanomaterials in the presence of overlapping resonances
LLinear and nonlinear optics of hybrid plasmon-exciton nanomaterials in the presenceof overlapping resonances
Maxim Sukharev ∗ Science and Mathematics Faculty, College of Letters and Sciences, Arizona State University, Mesa, Arizona 85212
Paul N. Day and Ruth Pachter † Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433 (Dated: August 20, 2018)We consider a hybrid plasmon-exciton system comprised of a resonant molecular subsystem andthree Au wires supporting a dipole mode which can be coupled to a dark mode in controllablefashion by variation of a symmetry parameter. The physics of such a system under strong couplingconditions is examined in detail. It is shown that if two wires supporting the dark mode are coveredwith molecular layers the system exhibits four resonant modes for a strong coupling regime due toasymmetry and lifted degeneracy of the molecular state in this case, while upon having molecularaggregates covering the top wire with dipolar mode, three resonant modes appear. Pump-probesimulations are performed to scrutinize the quantum dynamics and find possible ways to controlplasmon-exciton materials. It is demonstrated that one can design hybrid nanomaterials with highlypronounced Fano-type resonances when excited by femtosecond lasers. ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . op ti c s ] M a r I. INTRODUCTION
Understanding plasmon-exciton or so-called plexitonic coupling [1] in hybrid plasmonic nanostructures is importantfor tuning the optical response, e.g. for applications in nonlinear optics [2], organic solar cells [3], or organic light-emitting diodes [4]. In developing such nanostructures, it is important to consider strong coupling phenomena. Indeed,experimentally, Rabi oscillations were observed in J-aggregate/metal nanostructures [5–8]. For example, in the workby Schlather et al. [8], gold nanodisk dimers were utilized, with J-aggregates formed from monomers of a cyaninedye. Cynanine dyes were also used in the work by DeLacy et al. [5] with silver nanoplatelets. Limiting cases of Rabioscillations/Fano resonances were identified, where the plasmon resonance has an extremely narrow/large linewidth[9].Inducing a Rabi splitting, for example, could be useful in gaining increased tunability in the response. However,such tunability is limited, e.g. in varying the distance between dimers. Here we consider an asymmetric metalplasmonic nanostructure in conjunction with a quantum excitonic subsystem, and particularly the effects of theexcitonic subsytem on the coupling between the dipolar and quadrupole modes. We adopted the geometry similar tothat described by Gallinet et al. [10] by varying the symmetry parameter. In previous works, we considered plexitonicnanomaterials both in linear [11] and nonlinear [12, 13] regimes. It was shown that in addition to commonly observedRabi splittings, plexitonic systems exhibit collective resonances at high exciton densities. Such resonances correspondto collective electromagnetic modes induced by strong exciton-exciton interactions greatly enhanced by plasmons [11].The existence of plasmon-enhanced collective exciton modes have also been confirmed in core-shell materials [14].Moreover using the pump-probe technique one can modify optical properties of plexitonic structures via modificationsof exciton populations [12] and efficiently control electromagnetic localization both in space and time [13].Using a self-consistent rigorous approach, we show that upon strong coupling in the asymmetric structure similarto the one in Ref. [10], and with inclusion of a strongly coupled excitonic subsystem, also varying the position ofdeposition, interesting phenomena appear that have not been elucidated yet. Such systems improve on the tunability ofplexitonic hybrid materials. Furthermore, in performing pump-probe simulations we note appearance of a pronouncedFano lineshape.
II. THEORETICAL MODEL
A spatiotemporal dynamics of electric, (cid:126)E , and magnetic, (cid:126)H , fields is considered classically using the full machineryof the Maxwell equations µ ∂ (cid:126)H∂t = −∇ × (cid:126)E (1a) ε ∂ (cid:126)E∂t = ∇ × (cid:126)H − (cid:126)J, (1b)where ε and µ are the permittivity and the permeability of the free space, respectively, and (cid:126)J corresponds to eitherthe current density in spatial regions occupied by metal or the macroscopic polarization current, (cid:126)J = ∂ (cid:126)P∂t , in spacewith molecules. The equations (1) are integrated using the finite-difference time-domain (FDTD) approach, whichimplicitly accounts for Gauss’s law via Yee’s cell thus requiring only two equations to solve.To account for the material dispersion in a metal the Drude model is implemented. In this model the currentdensity is evaluated according to the following equation [15] ∂ (cid:126)J∂t + Γ (cid:126)J = ε ω p (cid:126)E, (2)where Γ is the damping parameter and ω p is the bulk plasma frequency. An additional parameter that enters theDrude model is the high-frequency limit of the dielectric constant ε r . For the range of frequencies considered in thiswork the following set of parameters was chosen to represent gold: ε r = 9 . ω p = 8 .
95 eV, and Γ = 0 .
069 eV. We notethat we tested a more precise description of the dielectric function of gold using the Drude-Lorentz model [16]. Theresults were qualitatively similar to those obtained with the Drude model, i.e. all resonances discussed below werepresent but their energy positions were slightly different. Moreover the physical nature of all resonances were the sameas we checked via calculating corresponding charge distributions. The Drude model however leads to significantlyshorter execution times of our codes hence it was used to capture essential physics rather than quantitative features.The optical response of a molecular aggregate is simulated using rate equations for a two-level system [17] drivenby a local electric field (cid:126)E dn dt − γ n = − (cid:126) Ω (cid:126)E · ∂ (cid:126)P∂t , (3a) dn dt + γ n = 1 (cid:126) Ω (cid:126)E · ∂ (cid:126)P∂t , (3b) ∂ (cid:126)P∂t + ( γ + 2 γ d ) ∂ (cid:126)P∂t + Ω (cid:126)P = − σ ( n − n (cid:126)E, (3c)where n and n describe the populations of the ground and the excited molecular states, respectively, (cid:126)P is themacroscopic polarization, γ is the radiationless decay rate of the excited state, γ d is the pure dephasing rate, and (cid:126) Ω is the energy separation of the molecular levels. The coupling constant σ can be derived from a simple harmonicoscillator as readily shown in Ref. [18] σ = 2Ω µ (cid:126) , (4)here µ is the transition dipole moment.The equations (1) and (3) are coupled via polarization current (cid:126)J = ∂ (cid:126)P∂t that appears in the Ampere law. Theresulting system of equations is solved numerically on a multi-processor computer. We note that in such an approachthe static molecule-molecule interactions are neglected thus allowing us to treat each molecule independently. Thismethod however does account for all interactions between molecules which are induced by external EM radiation.The values of the molecular parameters used in this work are: µ = 10 Debye, γ = 6 . × − eV (correspondingto 6 ps), γ d = 6 . × − eV (corresponding to 630 fs), and the total number density of molecules is n = 4 × m − . The thickness of a molecular layer in all simulations below is 10 nm. Finally we assume that the dielectric hostfor all structures considered in this work has a dielectric constant ε = 1 . δx = δy = δz = 1 . . . . δt = δx/ (2 c ) = 2 . c is the speed of light in vacuum. Open boundaries are simulated using convolutionalperfectly matched layers (CPML) [20]. We found that for systems considered here the best results were achieved with19 CPML layers.The excitation of a system is carried out using the total field/scattered field approach [19], which allows one toinject a plane wave into a simulation domain. For calculations of linear spectra we employ a short-pulse method [21]with a time pulse envelope, f ( t ), written in the form of the Blackman-Harris window f ( t ) = (cid:88) n = − a n cos (cid:18) πntτ (cid:19) (5)for a pulse with a duration τ , where a = 0 . a = − . a = 0 . a = − . z direction. Thesubsequent normalization of these calculations to the energy flux of the incident wave results in reflection as plottedin the main panel of Fig. 1.The upper wire acts as a dipole antenna coupling to incident radiation. The bottom two wires support a dark modewhich can not be directly excited by the external field. The dipole antenna however depending on the asymmetryparameter S (see the inset in Fig. 1) allows one to harvest the dark mode. The main panel shows the reflectioncalculated at a point near the system. It is seen from this figure that the more asymmetric the structure is thestronger the coupling between the dipole mode of the upper wire and the dark mode of the two parallel wires is. Thisobservation is in a good agreement with main results of Ref [10]. We note that the dipole mode for the upper wire is at1 . . ω [eV] r e f l ec ti on S = 0S = 5 nmS = 10 nmS = 15 nmS = 20 nmS = 25 nm
35 nm S
20 nm 60 nm 15 nm xy x z
FIG. 1. (Color online) The inset shows the arrangement of gold nanowires in a three-dimensional structure. The systemis excited by a plane wave polarized along the x -axis and propagating along the negative z direction. The fixed geometricalparameters such as the length of each wire, etc. are indicated in the inset. The main panel shows reflection spectra for thesystem as a function of the incident photon energy at different positions of the upper wire as indicated in the main panel legend.The vertical dashed line shows the position of the quadrupole mode at 1 . III. RESULTS AND DISCUSSION
The main goal of this work is to explore how the presence of resonant molecular aggregates influences couplingbetween bright and dark modes in the system shown in Fig. 1. We investigate two scenarios of a deposition ofmolecular aggregates. First ( geometry A ) is to cover the upper wire with a thin layer of resonant molecules to alterthe dipole mode which in turn changes the indirect coupling between the incident field and the dark mode of thesystem. The second scenario ( geometry B ) is to place molecules on a surface of two parallel lower wires in order to seeif a resonant molecular system can more efficiently change the coupling by interacting with the dark mode directly.Moreover the spatial proximity of resonant molecular layers in the case of geometry B should lead to a degeneratemolecular state and hence more than three modes in spectra at the strong coupling conditions.
A. Linear optical response
First we examine spectral features of the hybrid system (plasmons+excitons) in the linear regime calculatingreflection as a function of the incident photon energy. The linear regime in our calculations corresponds to molecularexcited state populations always much smaller than 1.Results for both molecular coverages are shown in Fig. 2 for the case of a molecular resonance centered at Ω = 1 . r e f l ec ti on S = 0S = 15 nmS = 20 nm w [eV]0.050.10.15 r e f l ec ti on (a)(b) FIG. 2. (Color online) Reflection spectra for two scenarios of a deposition of molecular aggregates. Panel (a) shows results assolid lines for the molecular layer covering the upper wire (see the inset in Fig. 1 for details). Panel (b) shows data as solidlines obtained with two lower wires covered by resonant molecules. In both scenarios the thickness of a molecular layer is 10nm with all molecules resonant at Ω = 1 . of the strong coupling between molecules and the dipole mode due to strong local field enhancement as it was initiallyanticipated. For the symmetric orientation of the upper wire, S = 0, one can see that the Rabi splitting of the dipoleresonance is about 97 meV. For asymmetric cases the splitting increases reaching the value of 158 meV at S = 20 nm.It is also clear from Fig. 2a that asymmetric arrangements of wires have three distinct resonant modes correspondingto two obvious hybrid exciton-plasmon modes due to the coupling between molecules and the dipole mode of theupper wire and the dark mode as expected.In comparing our results (Fig. 2) with the electromagnetic transparency window and energy storage results for theasymmetric structure in Ref. [10], we note a reduction in intensity for both the A and B coating suppositions in oursystem for S = 0, as expected, although larger than the reduction in intensity of more than 50% that was calculatedfor the strong coupling regime for the system in Ref. [10]. This has been rationalized in terms of the radiative andnonradiative decay contributions. It is interesting to note that in the case of S = 0, where the strong coupling withthe molecular layer induces a Rabi splitting, subsystem B demonstrates a blue-shift for the first peak as comparedto subsystem A and the splitting is less pronounced, as is expected. This introduces another parameter that couldenable modulation of the reflectance in these geometries.One may anticipate that for geometry B two lower wires covered with resonant molecules would exhibit additionalinteraction: molecular layer on wire 1 with molecular layer on wire 2. If the coupling between these layers and eitherthe dipole mode or the dark mode is strong the spectral response may show four resonances. The results for geometryB are shown in Fig. 2b. For the symmetric case, S = 0, we again observe the Rabi splitting but this time it isnoticeably smaller, 50 meV, compared to the previous case. This can be explained by the fact that the molecular e n e r gy [ e V ] W [eV] e n e r gy [ e V ] W [eV] (a) (b)(c) (d) FIG. 3. (Color online) Avoided crossings for two sets of geometries. Panels (a) and (b) show energy positions of all resolvedresonances in reflection spectra as functions of the molecular transition energy Ω for geometry A (molecular layer covers theupper wire only). Panels (c) and (d) show energies of all resonances for geometry B (molecular layer covers two lower parallelwires). The asymmetry parameter S = 0 for panels (a) and (c), and S = 20 nm for panels (b) and (d). Note that each resonanceis represented by circles with different colors. Horizontal dashed lines in each panel show corresponding non-interacting plasmonmodes. For the symmetric case S = 0 there is only one plasmon state while at S = 20 nm there are two states as discussed inFig. 1. layers now cover two lower wires and its EM coupling to the dipole mode supported by the upper wire is lower. Forasymmetric arrangements there are four clearly distinctive resonances observed. This indicates the strong couplingbetween all four subsystems: the dipole mode of the upper wire, the quadrupole mode of the two parallel lower wires,the molecular aggregate on wire 1, and the molecular aggregate on wire 2. The two last modes are degenerate withthe degeneracy lifted by their coupling with plasmon states of the three interacting wires.To further demonstrate the physics of such a system we performed a series of simulations varying the molecularresonant energy Ω for both geometries A and B and extracting energies of each mode for a given molecular transitionenergy. The results are presented in Fig. 3. Both geometries A and B with the symmetric arrangement S = 0 (Figs.3a and 3c) exhibit a single avoided crossing indicating the coupling between molecular aggregates and the dipole modesupported by the upper wire. Once the symmetry is broken ( S = 20 nm, Figs. 3b and 3d) a set of multiple avoidedcrossings appears. In the case of geometry A, where a single molecular state is coupled to both dipole-dark plasmonstates as Fig. 3b demonstrates with clearly distinct avoided crossings. Geometry B with a non-zero asymmetryparameter S exhibits more complex spectra (Fig. 3d). At low and high molecular transition energies both molecularlayers show a single resonance due to weak interaction between them. However when the molecular energy of bothlayers approaches the region of the strong coupling with hybrid plasmon states the degeneracy is removed and spectrashow four distinct resonances. It is interesting to note that when the molecular energy is about 1 .
57 eV (close to thedark plasmon mode) one of the molecular states is very narrow. The proximity of a narrow and a wide resonance
25 50 75 100time [fs]0.20.40.60.81 g r ound s t a t e popu l a ti on one emitterS = 0S = 15 nmS = 20 nm 1.3 1.4 1.5 1.6 1.7 1.8 ω [eV]0.10.20.30.4 r e f l ec ti on ∆τ = 0 ∆τ = 43 fs ∆τ = 100 fs1.3 1.4 1.5 1.6 1.7 1.8 ω [eV]0.0250.050.0750.1 r e f l ec ti on ∆τ = 0 ∆τ = 45 fs ∆τ = 100 fs 1.3 1.4 1.5 1.6 1.7 1.8 ω [eV]0.10.20.3 r e f l ec ti on ∆τ = 0 ∆τ = 41 fs ∆τ = 100 fs (a) (b)(c) (d) FIG. 4. (Color online) Pump-probe simulations for geometry A. Panel (a) shows time dynamics of the ground state population.Black lines show data for a single molecule in vacuum subject to the pump pulse. Red, green, and blue lines show ensembleaverage populations of the ground state as a function of time for S = 0, S = 15 nm, and S = 20 nm, respectively. Panel (b)shows reflection as a function of the probe photon energy with S = 0 at three pump-probe time delays: black line is for ∆ τ = 0(system is not altered by pump yet), red line is for ∆ τ = 43 fs (panel (b)), ∆ τ = 45 fs (panel (c)), ∆ τ = 41 fs (panel (d)), andblue line is for ∆ τ = 100 fs. Panels (c) and (d) show data similar to that presented in panel (b) but for S = 15 nm and S = 20nm, respectively. The pump pulse parameters are: pulse duration is 100 fs, peak amplitude is E = 2 . × V / m, centralfrequency is 1 .
57 eV. The molecular transition energy is 1 .
57 eV. could be used to coherently control EM energy distribution in such a system if properly altered by a strong externallaser pulse inducing the Fano-type interference.
B. Pump-probe simulations
We consider a pump-probe experiment during which a system under consideration is subject to strong externallaser radiation. After some time delay, ∆ τ , a low intense probe pulse arrives and ”measures” the corresponding linear response of the system altered by the pump. Numerical aspects of such simulations for hybrid plasmon-excitonsystems can be found in Ref. [12]. In all simulations presented below we define the pump-probe time delay such that∆ τ = 0 corresponds to the probe arriving just before the pump.First we examine geometry A pumping it with a 100 fs laser pulse with an amplitude corresponding to a π/
25 50 75 100time [fs]0.20.40.60.81 g r ound s t a t e popu l a ti on one emitterS = 0S = 15 nmS = 20 nm 1.3 1.4 1.5 1.6 1.7 1.8 w [eV]0.050.10.15 r e f l ec ti on Dt = 0 Dt = 62 fs Dt = 100 fs1.3 1.4 1.5 1.6 1.7 1.8 w [eV]00.030.060.090.12 r e f l ec ti on Dt = 0 Dt = 52 fs Dt = 100 fs 1.3 1.4 1.5 1.6 1.7 1.8 w [eV]00.020.040.060.08 r e f l ec ti on Dt = 0 Dt = 52 fs Dt = 100 fs (a) (b)(c) (d) FIG. 5. (Color online) Same as in Fig. 4 but for geometry B. Parameters of the molecular system and the pump are the same. observed faster dynamics can be explained by local field enhancement due to the excitation of a corresponding plasmonmode. Higher local field increases the pump pulse area leading to more Rabi cycles. Another important aspect topoint out is a quick decay of the amplitude of Rabi oscillations. Upon careful examination of spatial distributions ofexcited molecules one can conclude that the excitation travels fast spreading over the entire molecular layer. Since thelayer is not uniformly exposed to the pump this excitation is highly inhomogeneous leading to a quick dephasing ofmacroscopic polarization currents [6]. Another evidence to support this explanation is that the Rabi splitting at theend of the pump is considerably lower compared to its unperturbed value (for S = 0 the splitting is 94 meV for theunperturbed spectrum and is only 42 meV at the end of the pump). The excitation spreads quickly over the entiremolecular layer thus leading to a high dephasing which in turn lowers the coupling with the hybrid plasmon modeand hence affects the Rabi splitting.To scrutinize the pump dynamics further we probe the perturbed system at different times as shown in Figs. 4b, 4c,and 4d. The system undergoes a strong transition near 40 −
50 fs as seen in Fig. 4a as a deep minimum of the groundstate population, which reaches almost 20%. Probing the system at these times reveals that the molecular aggregatetransitions into excited state. The reflection spectra calculated around ∆ τ = 40 fs exhibit a narrow resonance at themolecular transition energy 1 .
57 eV as the system releases the energy it stored due to the pump. Probed at latertimes the system shows less and less transitions occurring at 1 .
57 eV as the excitation quickly dephases as seen fromblue lines in Figs. 4b, 4c, and 4d.Completely different physics is seen when examining geometry B as shown in Fig. 5. Firstly the quantum dynamicsof the molecular subsystem (Fig. 5a) is noticeably slower compared to its counterpart (Fig. 4a). The latter showsthat all arrangements with different asymmetry parameters result in similar time dynamics. On the contrary, thesymmetric case with S = 0 for geometry B is considerably slower than others. Probing the system with S = 0 atdifferent time delays ∆ τ shows basically the same trend we noted above, i.e. at the time when the average groundstate population has a minimum the reflection spectrum has a sharp resonance indicating the fact that a portion of x [nm] z [ n m ] −
50 0 50 − x [nm] y [ n m ] −
50 0 50 − x [nm] y [ n m ] −
50 0 50 − − (c) (d)(b)(a) FIG. 6. (Color online) Steady-state electromagnetic intensity distributions calculated for ω = 1 . x andpropagates in the negative z -direction. molecules is in the excited state. It again decays very quickly and relaxes back to the unperturbed spectrum due toquick dephasing. However for all asymmetric cases with S = 15 nm (Fig. 5c) and S = 20 nm (Fig. 5d) we observethe strong induced Fano-type resonance with the reflection becoming negligibly small at 1 .
56 eV. As we anticipated inSection III A while discussing avoided crossings the proximity of sharp and broad overlapping resonances may lead tostrong interference resulting in Fano lineshapes. Here the interference is caused by the excitation of one of the hybridmolecular-plasmon states and its strong coupling with another hybrid state. We note that such a highly pronouncedFano lineshape with a deep minimum in the reflection leads to high local field enhancement if being excited at thefrequency corresponding to this minimum [22].To illustrate how the system behaves at frequencies corresponding to very small reflection as seen in Fig. 5c and5d we calculate steady-state electromagnetic energy distributions as shown in Fig. 6. First the structure is pumpedby a femtosecond pulse and then probed with a time delay of ∆ τ = 52 fs at the frequency of ω = 1 . IV. CONCLUSION
We scrutinized optical properties of a hybrid plasmon-exciton system comprised of a resonant molecular subsystemand three Au wires supporting a dipole mode which can be coupled to a dark mode in controllable fashion by variationof a symmetry parameter. It was shown that if two wires supporting the dark mode are covered with molecular layersthe system exhibits four resonant modes for a strong coupling due to asymmetry and lifted degeneracy of the molecularstate in this case, while upon having molecular aggregates covering the top wire with dipolar mode, three resonantmodes appear. Furthermore we used pump-probe simulations to study the quantum dynamics and possible ways0to control plasmon-exciton materials. It was demonstrated that one can design hybrid nanomaterials with highlypronounced Fano-type resonances using femtosecond lasers. We showed that one can use ultra-short laser pulses todesign very efficient nano-absorbers. The femtosecond optical engineering of plexitonic systems can be used to creatematerials with desired properties and functionality.
ACKNOWLEDGEMENTS
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