Anisotropic Optical Response of Nanostructures with Balanced Gain

Article pubs.acs.org/journal/apchd5

Anisotropic Optical Response of Nanostructures with Balanced Gain and Loss Alejandro Manjavacas* Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, United States S Supporting Information *

ABSTRACT: Photonic systems containing active and passive elements with balanced gain and loss are attracting increased attention due to their extraordinary properties. These structures, usually known as 7; -symmetric systems, display strongly asymmetric behaviors. Here we study the optical response of finite nanostructures composed of pairs of active and passive nanospheres operating close to the 7; -symmetry condition. We find that, despite their highly regular geometry, these systems scatter light predominantly toward the gain side of the structure when illuminated perpendicularly to their axis. Furthermore, the backscattering intensity for illumination parallel to the axis depends strongly on the side of incidence, being several times larger for light coming along the gain side. Interestingly, under the same conditions, the forward scattering and, consequently, the extinction cross-section remain independent of the side of incidence. This leads to an asymmetric absorption cross-section that can be made arbitrarily small for light impinging on the gain side of the structure. These results contribute to the basic understanding of the optical properties of active−passive finite nanostructures with potential applications for the design of novel nanostructures displaying asymmetric and tunable responses. KEYWORDS: nanoparticles, gain, PT-symmetry, scattering, absorption, asymmetric

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directly translated into a requirement for the dielectric function of the system, ε(r) = ε*(−r), which can be achieved using active elements.31,32 The interest in 7; -symmetric photonic systems relies on their exceptional properties including, to cite some, asymmetric propagation,33,34 reflection,35 scattering,36 unidirectional invisibility,37,38 switching,39−41 and extraordinary nonlinear behaviors,42−44 which have been already observed in dielectric waveguides34,45−47 and cavities48,49 or photonic lattices,50,51 among other realizations. More recently, 7; -symmetric systems built using plasmonic elements have attracted increased attention. The reason is that the strong interaction of surface plasmons with light can be exploited to enhance the extraordinary properties arising from the 7; -symmetry.52 Recent works along these lines have explored asymmetric behaviors in waveguides53 and metamaterials,54,55 transition from absorption to amplification in cavities56 and waveguides,57 unidirectional cloaking,58 switching,59 multiplexing,60 anisotropic emission in hybrid nanoparticles,61 and giant near-field radiative heat transfer.62 In this paper we investigate the optical response of finite plasmonic systems operating near the 7; -symmetry condition with the aim of finding strongly asymmetric behaviors. In order to do so, we focus on structures composed of pairs of active and passive nanospheres, whose response is analyzed, using realistic

anostructures made of metallic materials are well known to support collective oscillations of the conduction electrons, commonly referred to as surface plasmons.1 These excitations couple strongly with light, producing large interaction cross-sections and near-fields.2 Such extraordinary properties have been already exploited in a variety of applications including photothermal cancer therapy,3 ultrasensitive biosensing,4 and improved solar energy harvesting,5 among others. However, the strong fields and large crosssections produced by surface plasmons come at the price of relatively high levels of losses.6 One way of fighting these losses is to use active materials such as dye molecules, rare earth ions, or semiconductor crystals, which can be externally pumped, thus providing gain to the system and consequently reducing its intrinsic losses.7−10 Interestingly, the presence of gain in a plasmonic system not only serves to mitigate the losses11−14 but also enables the development of novel light-emitting devices,15,16 such as spasers17−20 or, more generally, plasmonic nanolasers.21−27 It is also important to note that the possibility of tuning the level of gain provides a path to actively control the response of these systems.28,29 In the past years, a new wave of interest in active materials has come associated with the proposal of 7; -symmetric systems.30 These systems consist of arrangements of active and passive elements, such that they present a balanced gain and loss. Photonic platforms are ideally suited to implement these structures since, in that case, the 7; -symmetry condition is © 2016 American Chemical Society

Received: April 16, 2016 Published: June 17, 2016 1301

DOI: 10.1021/acsphotonics.6b00274 ACS Photonics 2016, 3, 1301−1307

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Figure 1. Optical response of a 7; -dimer. (a) Schematics of the system under study consisting of two spherical nanoparticles of radius R separated by a distance d, which are illuminated with a plane wave propagating and polarized perpendicularly to the dimer axis. One of the particles displays optical gain (red), while the other one has a passive response (blue). (b) Extinction cross-section for a dimer with R = 50 nm and d = 125 nm as a function of the pumping parameter F, and the detuning with respect to the gain frequency ω0. The red dashed curve signals the regions for which the extinction cross-section is larger than 10 times the cross-sectional area 2πR2. (c) Extinction (black dashed curves), scattering (red solid curves), and absorption (blue solid curves) cross-sections for the same dimer of panel b calculated for three different pumping levels: F = 0 (bottom), F = 0.55 (middle), and F = 0.6 (top). In all cases the cross-sections are normalized to the cross-sectional area of the dimer.

dielectric functions, by solving Maxwell’s equations with the help of a multiple elastic scattering of multipolar expansions (MESME) method.63,64 We find that an active−passive dimer illuminated with light propagating and polarized perpendicularly to its axis scatters that light predominantly along the direction of the gain side of the structure. The anisotropy in the scattered light is more pronounced when the number of active−passive dimers in the system is increased. A similar asymmetric behavior is predicted for illumination parallel to the axis of the structure. In that case, the scattering intensity depends strongly on the side of incidence, with the backscattering being several times larger for light coming along the gain side. Interestingly, under the same conditions the forward scattering is always independent of the side of incidence, and therefore so is the extinction cross-section. This produces an anisotropic absorption cross-section that can be made arbitrarily small for light impinging along the gain side of the system. The results presented in this work help to understand the behavior of finite active−passive nanostructures operating close to the 7; -symmetry condition and pave the way to design novel photonic systems displaying strongly asymmetric and tunable responses.

This term describes the gain produced by optically pumped quantum emitters such as dye molecules or rare earth ions.61,62 We chose a gain frequency ℏω0 = 1 eV and a gain bandwidth γ0 = 0.01ω0, which are similar to the values used in other studies.61,62 On the other hand F, which is proportional to the population inversion of the gain media, is left as a parameter that controls the pumping level and hence the amount of gain in the system. Figure 1b shows the extinction cross-section of the dimer, normalized to its cross-sectional area, 2πR2, calculated for different values of F. The external illumination is assumed to propagate and be polarized perpendicularly to the nanostructure axis, as shown in Figure 1a. For a passive dimer (i.e., F = 0) the extinction cross-section is almost uniform over the explored frequency range, which corresponds to the dipolar plasmon of the dimer.66 As the pumping increases, the gain starts to compensate the losses of the system, and a strong peak emerges near ω0 with values rapidly rising above 10 (red dashed curve). However, at some critical pumping value (F = 0.612 for this system), the peak disappears and the extinction cross-section vanishes (white area). Beyond that point, a region with negative values appears (blue area). Interestingly, other critical points, associated with higher order modes of the dimer, emerge at higher values of F. We explore these different behaviors in more detail in Figure 1c, where the extinction (black dashed curves), scattering (red solid curves), and absorption (blue solid curves) cross-sections of the dimer are plotted for three different values of the pumping parameter. As anticipated, when F = 0 (bottom panel), we observe a resonance that corresponds to the dipolar mode of the dimer, for which the extinction cross-section is several times larger than 2πR2. As the pumping increases to F = 0.55 (middle panel), the gain mitigates some the absorption losses. This produces a narrower and stronger resonance for the extinction and the scattering cross-sections, but also a more uniform absorption. Close to the critical point, F = 0.6 (top panel), the gain has completely overcome the losses and the absorption cross-section becomes negative, displaying a very sharp resonance that almost compensates the scattering crosssection, thus yielding a small extinction. This is the signature of an overall gain in the nanostructure and also indicates the proximity of a lasing threshold. It is important to notice that the

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RESULTS AND DISCUSSION We start our study by considering a dimer composed of two nanospheres of radius R = 50 nm, separated by a distance d = 125 nm, as shown in Figure 1a. One of the particles (blue) has a passive metallic response that is modeled using a Drude dielectric function:1 εL(ω) = ε∞ −

ωp2 ω 2 + iωγp

with ε∞ = 2, ℏωp = 2 eV, and γp = 0.05ωp. These values are a realistic approximation for transparent conductive oxides at frequencies below their band gap.65 The other particle in the dimer (red) has an active response that we describe by adding a Lorentzian gain term to the dielectric function: γ0 εG(ω) = εL(ω) + F ω − ω0 + iγ0 1302

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symmetrical for a passive dimer (i.e., F = 0). However, as the pumping parameter increases, and therefore the system approaches the 7; -symmetry condition, the scattering becomes more and more asymmetric (cf. red and blue curves), which is reflected by the ratio of the intensities scattered along the two directions (black dashed curve) growing from 1 to a value close to 2 for F = 0.55. At that point, as shown in panel c, the angular distribution of the scattering produced by the active−passive dimer (red solid curve) completely differs from that of a passive dimer (black dashed curve). Not only it is much larger due to the gain (note the scaling factor), but it also displays a strong imbalance toward the gain side of the dimer. This is somehow surprising since, given the small values of d and R as compared with the wavelength (∼1240 nm), we would expect the dimer to behave effectively as a dipole, producing a completely symmetrical radiation pattern, which for the configuration considered here would be ∝1 + cos2 θ. Indeed, in the dipolar limit, the power scattered by two nanoparticles with polarizabilities αG and αL, illuminated perpendicularly to the axis connecting their centers, is given by (see Methods)

value of F at the critical point depends on the concentration and the properties of the gain media as well as on the geometry of the structure. The sharp changes occurring in the optical response of the dimer as F approaches the critical point are a typical signature from 7; -symmetric systems.45,53 Therefore, associated with that behavior, we expect the system to exhibit an anisotropic response. In order to explore such a possibility, we analyze in Figure 2a the intensity of the light scattered by the dimer of Figure 1 along its axis, in the direction of both the gain (θ = 0°, red solid curve) and the loss (θ = 180°, blue solid curve) sides (see the inset). We compute this as a function of F for values not larger than 0.55, for which the absorption remains positive, thus ensuring we are far from any lasing threshold, and for a detuning ω − ω0 = −0.18γ0. As expected, the scattering is

P(θ ) =

ω4(1 + cos2 θ ) [|αG|2 + |αL|2 2c 3 + 2Re{e−ikdcosθ αGαL*}]

(1)

where k = ω/c and θ is the angle between the scattering direction and the dimer axis (see the inset in Figure 2a). Notice that we assume a unit incident field and use Gaussian units. The ratio of the scattering intensities along the θ = 0° and θ = 180° directions can be written as (1 + β)/(1 − β) with β=

2 sin(kd) Im{αGαL*} 2

|αG| + |αL|2 + 2 cos(kd) Re{αGαL*}

(2)

Clearly, β = 0 for passive dimers (αG = αL). This is also the case even if αG = AαL, as long as A is a real number. Actually, the only way of obtaining an asymmetric behavior (i.e., β ≠ 0) is by having a phase difference between αG and αL so that Im{αGα*L } ≠ 0. This is, indeed, what happens when the dimer approaches the 7; -condition, for which αG = αL*. The dependence of the asymmetric behavior on the geometrical parameters R and d as well as in the material losses γp is explored in detail in Figures S1 and S2 of the Supporting Information. In particular, the increase in the separation between the particles or their radius results in larger asymmetries in the scattering distribution but at the price of reducing the intensity of the scattered light. Interestingly, the asymmetry in the scattering pattern can be enhanced by adding more active−passive dimers to the system. This is illustrated in Figure 2c and e, where we repeat the calculations of panel a but for a tetramer and a hexamer, as shown in the corresponding insets. In these cases the ratio of the scattering intensities along the θ = 0° and θ = 180° directions is increased to ∼5 for the tetramer and ∼8 for the hexamer, for values of the pumping parameter F = 0.57 and F = 0.59, and detunings ω − ω0 = −0.41γ0 and ω − ω0 = −0.35γ0, respectively. As expected, the corresponding angular distributions of the scattered light, shown in panels d and f, are strongly biased toward the side of the structure with gain, which is in sharp contrast to the symmetrical distributions displayed by their passive counterparts (cf. red solid and black dashed curves).

Figure 2. Anisotropic scattering produced by 7; -symmetric nanostructures. (a) Scattering intensity produced by a dimer along its axis in the direction of the gain side θ = 0° (red solid curve) and the loss side θ = 180° (blue solid curve) as a function of the pumping parameter F and for a detuning ω − ω0 = −0.18γ0. The black dashed line (right scale) shows the ratio between the two scattering intensities. (b) Angular distribution of the scattering intensity for F = 0.55 (red solid curve) compared with that of the passive dimer F = 0 (black dashed curve). The latter is multiplied by 5 to improve the visibility. (c−f) Same as panels a and b but for a tetramer (ω − ω0 = −0.41γ0) (c, d) and for a hexamer (ω − ω0 = −0.35γ0) (e, f). In all cases, the particles have a radius R = 50 nm and are separated by a distance d = 125 nm. Furthermore, the scattering intensities are normalized to the corresponding cross-sectional areas, and the external illumination propagates and is polarized perpendicularly to the nanostructure axis, as shown in the insets of panels a, c, and e. 1303

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backscattering is larger when light is incident on the gain side (red solid curves) than on the loss side (blue solid curves). However, as anticipated, the forward scattering (black dashed curves) remains equal for these two situations. The asymmetry in the backscattering increases as the pumping becomes larger and the system approaches the 7; -symmetry condition. This behavior is analogous to the asymmetric transmission and reflection found in extended 7; -symmetric systems.37,54 Figure 3b delves deeper into this asymmetry by showing the ratio of the scattering intensity for incidence along the gain and loss sides of the nanostructure as a function of the scattering angle θ. The red curve corresponds to the dimer, while the blue one represents the tetramer. In both cases we choose F = 0.53, which coincides with the maximum F considered in panel a and for which both nanostructures exhibit positive absorption. We see that for θ = 0° the ratio reaches values of ∼4 for the dimer and ∼7 for the tetramer and decreases to 1 when θ = −180° or θ = 180°. The symmetrical behavior of the forward scattering intensity implies, by virtue of the optical theorem,67 the invariance of extinction cross-section with respect to the side of incidence. Furthermore, since the latter is the sum of the scattering and the absorption cross-sections, any change in the scattering associated with the illumination scheme has to come together with the opposite change in the absorption cross-section. As a consequence of this, the structures under study also display an asymmetric absorption cross-section. We explore such behavior in Figure 4a, where we plot the extinction (dotted curves), scattering (dashed curves), and absorption (solid curves) cross-

Another interesting situation appears when the incident light propagates parallel to the dimer axis (i.e., along the z-axis of Figure 1a). In that case, working again in the dipolar limit, and assuming that the field is incident in the gain side of the dimer, we find that the forward scattering intensity can be written as (see Methods) Pf =

ω4 [|αG|2 + |αL|2 + 2Re{αGαL*}] c3

(3)

Notice that this expression is invariant under the interchange of the subscripts G ↔ L, which means that the forward scattering is independent of the side of incidence of the external light. However, this is not true for the backscattering, which, for the same conditions, reads (see Methods) Pb =

ω4 [|αG|2 + |αL|2 + 2 cos(2kd) Re{αGαL*} c3 + 2 sin(2kd) Im{αGαL*}]

(4)

In this case, the interchange of G ↔ L flips the sign of the last term, which is proportional to Im{αGαL*}, as was β (see eq 2). Therefore, we expect an active−passive dimer to exhibit a different backscattering intensity depending on whether light is incident on the gain or the loss side of the nanostructure. This is illustrated in Figure 3a for a dimer (left panel) and a tetramer (right panel). There we plot the forward and the backscattering intensities as a function of the pumping parameter for ω − ω0 = −0.18γ0 and ω − ω0 = −0.23γ0, respectively. In both cases, the

Figure 3. Asymmetric backscattering produced by 7; -symmetric nanostructures. (a) Scattering intensity as a function of the pumping parameter F for a dimer (left panel) and for a tetramer (right panel), both composed of particles with R = 50 nm separated by a distance d = 125 nm. The red (blue) curves show the backscattering intensity for light impinging on the gain (loss) side of the nanostructure, while the black dashed curves stand for the forward scattering intensity, which is exactly the same for the two illumination configurations (see the main text). The detuning for the dimer is ω − ω0 = −0.18γ0, while for the tetramer it is ω − ω0 = −0.23γ0. The scattering intensities are normalized to the corresponding cross-sectional areas. (b) Ratio between the scattering intensities produced for incidence along the gain and the loss sides calculated for different scattering angles (see the inset). The red curve shows the results for the dimer, and the blue one for the tetramer. In both cases F = 0.53, and the detunings are the same as in panel a.

Figure 4. Anisotropic absorption produced by 7; -symmetric nanostructures. (a) Extinction (dotted curves), scattering (dashed curves), and absorption (solid curves) cross-sections for a dimer with R = 50 nm and d = 125 nm as a function of the pumping parameter F. The red (blue) curves show the results for light impinging on the gain (loss) side of the nanostructure. (b) Absorption cross-section density as a function of the position along the dimer axis for F = 0.538. The red (blue) curve shows the results for the case of light incident on the gain (loss) side of the dimer. The different numbers in the figure represent the absorption cross-section obtained by integrating the corresponding parts of the curves. In both panels the detuning is chosen to be (ω − ω0)/γ0 = −0.29. 1304

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sections for a dimer with R = 50 nm and d = 125 nm as a function of F. We perform these calculations for light incident on the gain (red curves) or loss (blue curves) sides of the system (see inset) and for ω − ω0 = −0.29γ0. As expected, while the extinction is the same in both situations, the absorption cross-section displays an asymmetric behavior that increases with F. Indeed, this quantity can be made arbitrarily small for incidence along the gain side by appropriately tuning the pumping parameter F, while, at the same time, it takes large values for the opposite illumination configuration (cf. red and blue solid curves). From a physical point of view, what happens at this point is that the emission in the active element of the dimer balances the absorption in the passive particle, thus resulting in a vanishing net absorption. This is illustrated in Figure 4b, which shows the absorption cross-section density as a function of the position along the axis of the dimer for F = 0.538. Clearly, when light impinges on the loss side (blue curve), there is an imbalance between the gain and the loss in the particles; the normalized absorption cross-section in the passive particle, obtained from the integration of the corresponding part of the curve, is 43.6, while in the active one it is −30.2, both in units of the cross-sectional area. However, when light is incident on the gain side (red curve), the total area under the curve cancels out, thus resulting in a negligible absorption cross-section. Therefore, under these conditions, the dimer behaves as a strongly anisotropic absorber.

to externally control the optical properties of the structure. The results presented here advance our understanding of the response of finite active−passive nanostructures operating near the 7; -symmetry condition, thus opening new avenues for the design of novel nanostructures with asymmetric and tunable responses.

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METHODS Dipolar Approximation. The power scattered by two particles per unit of solid angle can be written, in the dipolar limit (using Gaussian units), as dP ω4 = dΩ 2πc 3

∑

[|pi|2 − |pi ·n|2 ] +

i = G,L

ω4 πc

× Re{e−ikdcosθ [pG ·p*L − (pG ·n)(p*L ·n)]}

Here k = ω/c is the wavenumber, d = rL − rG is the vector connecting the particle centers, n is the unit vector pointing in the scattering direction, θ is the angle between d and n, and pG and pL are the dipole moments of the particles. These can be written in terms of the particle polarizabilities as pG = αGEG and pL = αLEL, where EG,L is the amplitude of the field at the position of the particles. It is important to note that, in all cases, we chose the field to be polarized perpendicularly to d and that αL and αG are the effective polarizabilities that already include the effect of the interaction between the two particles. When the field propagates perpendicularly to d, we have that EG = EL = E, and therefore

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CONCLUSIONS In summary, we have studied the optical response of nanostructures composed of pairs of active and passive nanospheres operating near the 7; -symmetry condition. By rigorously solving Maxwell’s equations, using realistic dielectric functions, we have demonstrated that these systems present a strongly asymmetric response. In particular, they scatter light predominantly along the gain side of the structure when illuminated perpendicularly to their axis. This asymmetric behavior is more pronounced when the number of active− passive dimers in the system is increased. Further, when the illumination is parallel to the axis, the backscattering intensity produced by these structures depends strongly on the incidence side, being several times larger for light impinging on the gain side of the system. However, at the same time, the forward scattering and, therefore, the extinction cross-section remain invariant. As a result of this, the system displays an asymmetric absorption cross-section that can be made arbitrarily small for incidence along the gain side, remaining finite for the opposite illumination configuration. It is important to note that we have analyzed structures composed of homogeneous particles; however other geometries might be more convenient for an experimental implementation, such as nanoshells with a plasmonic core and a gain/loss shell, similar to those commonly used in the design of spasers.18,20 Furthermore, all of our calculations have been performed assuming a homogeneous gain profile. However, inhomogeneities arising for large active particles can generate additional anisotropic behaviors.61 The active− passive dimers studied here can be employed as the basic building blocks to construct metamaterials with strongly asymmetric response. For example, the asymmetry in the absorption cross-section can be exploited to design highly directional absorbers. Additionally, the dependence of the system response on the level of pumping provides a mechanism

dP ω4 [|αG|2 + |αL|2 + 2Re{e−ikdcosθ αGαL*}] = 3 dΩ 2πc ×(|E|2 − |n ·E|2 )

which upon integration over the azimuthal angle and taking |E|2 = 1, leads to eq 1. On the other hand, when the field propagates parallel to d and is incident on the gain side, we have that EG = E and EL = E eikd, and therefore dP ω4 [|αG|2 + |αL|2 + 2Re{e−ikdcos θ e−ikdαGαL*}] = dΩ 2πc 3 ×(|E|2 − |n ·E|2 )

Again, integrating this expression over the azimuthal angle and taking |E|2 = 1 we obtain eqs 3 and 4. Full Solution of Maxwell’s Equations. All the results shown in the different figures have been obtained by solving Maxwell’s equations using a semianalytical approach based on the MESME method.63,64 The accuracy of these calculations has been verified by comparing them with equivalent results obtained using the boundary element method (BEM).68

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.6b00274. Additional plots similar to Figure 2a for different combinations of R and d and for different values of γp (PDF) 1305

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge financial support from the Department of Physics and Astronomy and the College of Arts and Sciences of the University of New Mexico, and the UNM Center for Advanced Research Computing for computational resources used in this work. We are also grateful to Prof. Javier Garcı ́a de Abajo, Prof. Peter Nordlander, and Ms. Lauren Zundel for valuable and enjoyable discussions and for their critical reading of the manuscript.

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