Giant frequency-selective near-field energy transfer in active--passive structures

Giant frequency-selective near-field energy transfer in active–passive structures

Chinmay Khandekar, Weiliang Jin, Owen D. Miller, Adi Pick, and Alejandro W. Rodriguez

Phys. Rev. B 94, 115402 – Published 1 September 2016

Abstract

We apply a fluctuation electrodynamics framework in combination with semianalytical (dipolar) approximations to study amplified spontaneous energy transfer (ASET) between active and passive bodies. We consider near-field energy transfer between semi-infinite planar media and spherical structures (dimers and lattices) subject to gain, and show that the combination of loss compensation and near-field enhancement (achieved by the proximity, enhanced interactions, and tuning of subwavelength resonances) in these structures can result in orders of magnitude ASET enhancements below the lasing threshold. We examine various possible geometric configurations, including realistic materials, and describe optimal conditions for enhancing ASET, showing that the latter depends sensitively on both geometry and gain, enabling efficient and tunable gain-assisted energy extraction from structured surfaces.

Received 13 November 2015
Revised 1 August 2016

DOI:https://doi.org/10.1103/PhysRevB.94.115402

Physics Subject Headings (PhySH)

Fluctuation theorems PT-symmetric quantum mechanics Spontaneous emission Thermodynamics

Dipole approximation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Chinmay Khandekar¹, Weiliang Jin¹, Owen D. Miller², Adi Pick³, and Alejandro W. Rodriguez¹

¹Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
²Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
³Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

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Vol. 94, Iss. 11 — 15 September 2016

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Images

Figure 1
Schematic of two semi-infinite plates of permittivities $ε_{1}$ and $ε_{2}$ , respectively, separated by a vacuum gap $d$ . Fourier decomposition of scattered waves with respect to parallel $k_{∥}$ and perpendicular $γ$ wave vectors simplifies calculations of energy transfer.
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Figure 2
Schematic of dimer system consisting of two spheres of permittivities $ε_{1}$ and $ε_{2}$ and radii $R_{1}$ and $R_{2}$ , respectively, and separated by a gap $d$ . Mie-series decomposition of scattered fields simplifies calculations of energy transfer; shown are a flux evaluation point $x = x_{1} = x_{2}$ in medium 0, with $x_{i}$ , denoting the position relative to the center of sphere $i$ .
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Figure 3
Far-field flux $Φ_{0} (ω)$ and flux-transfer $Φ_{12} (ω)$ associated with a dimer of two spheres of equal radii $R$ , permittivities $ε_{1} = ε_{r} + ε_{G}$ and $ε_{2} = ε_{r}$ , with $I m ε_{r} = 0.05$ , and separated by distance of separation $d$ , under various operating conditions. (a) Dependence of $Φ_{0} (ω)$ and $Φ_{12} (ω)$ on $I m ε_{1} < 0$ (under gain) at fixed $R e ε_{1, 2} = - 1.522$ , and for either $d \to \infty$ (left) or $d / R = 0.3$ (middle/right). White circles indicate the lasing threshold of a few individual modes while white dashed lines indicate operating parameters (cross sections) for the plots in (b), which show $Φ_{12}$ (solid lines) and $Φ_{0}$ (dashed lines) at fixed $I m ε_{1} = - I m ε_{2} = - 0.05$ and $d / R = 0.3$ . The plots compare the flux rates of gain–loss (GL) dimers (red lines) against those of passive (LL) dimers (blue lines).
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Figure 4
(a) Flux-transfer rate $Φ_{12}$ associated with the sphere dimer system of Fig. 3 under a simple dipolar approximation (DA), in either passive ( $I m ε_{1, 2} = 0.01$ , top) or active ( $I m ε_{1} = - I m ε_{2} = - 0.1$ , bottom) regimes, as a function of $R e ε_{1, 2}$ and $d / R$ . While the flux rate diverges in the active case under total loss compensation, only the rate per unit volume diverges in the case of finite, passive spheres. The validity of the DA for large $d > R$ is illustrated in (b), which shows also results obtained using the semianalytical (SA) equations [(9) and (10)]. (c) Flux rate spectra $Φ_{0} (ω)$ (top) and $Φ_{12} (ω)$ (bottom) of the dimer system under the $PT$ symmetry condition, $R e ε_{1, 2} = - 1.522$ and $I m ε_{1} = - I m ε_{2} = - 0.05$ , illustrating the splitting of a subwavelength dimer mode as $d$ changes around a critical $d_{c} \approx 0.306 R$ . The two branches include both quasistatic $ω_{0}^{(-)}$ and subwavelength $ω_{0}^{(+)}$ resonances. (d) Flux spectra at three different separations $d \approx {0.3056, 0.302, 0.3017} R$ , marked by the white dots (i), (ii), and (iii), respectively, in the bottom contour in (c).
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Figure 5
(b) Flux transfer $| Φ_{12} | R^{2} / A$ and (c) far-field flux $| Φ_{0} | R^{2} / A$ associated with the system shown schematically in (a), involving an infinite, two-dimensional lattices of gain and loss spheres of equal radii $R$ and period $t \approx d$ , and separated by a (varying) vertical distance $d$ , for different choices of $t / R ≳ 1$ and fixed values of $R e ε = - 1.95$ and $I m ε_{1} = - I m ε_{2} = - 0.01$ . Also shown are the corresponding flux rates obtained using a simple pairwise approximation (PA, dashed red lines) that ignores multiple scattering (see text), or associated with either passive spheres (LL, black solid lines) or isolated dimers (blue lines, both DA and SA). The flux rates are normalized by either the dimensionless unit areas $A / R^{2}$ in the case of lattices, with $A = {(t + 2 R)}^{2}$ , or $A = 4 π R^{2}$ in the case of an isolated dimer. (d) Compares the maximum achievable flux rate $| Φ_{12} | R^{2} / A$ in sphere lattice (solid lines) versus planar (dashed lines) geometries as a function of the ratio $I m ε_{1} / I m ε_{2}$ (relative overall permittivity of the gain spheres/plates) for two different choices of $I m ε_{2} = {0.01, 0.1}$ (red, blue) and fixed lattice parameters $d / R = t / R = 2$ .
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Figure 6
(a) Far-field flux $Φ_{0} (ω)$ (blue line) and flux-transfer $Φ_{12} (ω)$ (red line) spectra of a dimer consisting of two Ga-doped zinc-oxide spheres of radii $R = 0.2 c / ω_{21}$ , separated by a distance $d / R = 0.5$ . One of the spheres is doped with chromium ( ${Cr}^{2 +}$ ) ions having transition wavelength $λ_{21} = 2.51 μ m$ , and pumped to a population inversion $D_{0} = 0.375 (ℏ γ_{⊥} / 4 π^{2} g^{2})$ . Also shown is the far-field emission $Φ_{0} (d \to \infty)$ of the isolated gain sphere (green line). The top inset shows the peak ratio $Φ_{12}^{max} / Φ_{0}^{max}$ with respect to changes in $R$ , keeping $d / R$ and $D_{0}$ fixed. (b) Contour plots illustrating variations in $Φ_{0}$ (left/middle) and $Φ_{12}$ (right) with respect to $D_{0}$ , with the black dashed lines indicating operating parameters in (a). (c) Maximum spectral flux rates $| Φ_{12} (ω) | R^{2} / A$ (left) and $| Φ_{0} (ω) | R^{2} / A$ (right) for extended sphere lattices comprising GZO gain–loss spheres operating at $D_{0} = 0.3 (ℏ γ_{⊥} / 4 π^{2} g^{2})$ , well below the LT, but of radii $R \sim 0.05 c / ω_{21}$ , as a function of $d / R$ and for different values of $t / R$ . Also shown are the flux rates of passive lattices (LL, black solid lines), obtained by letting $D_{0} = 0$ .
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