Fluctuating volume-current formulation of electromagnetic fluctuations in inhomogeneous media: Incandescence and luminescence in arbitrary geometries

Athanasios G. Polimeridis, M. T. H. Reid, Weiliang Jin, Steven G. Johnson, Jacob K. White, and Alejandro W. Rodriguez

Phys. Rev. B 92, 134202 – Published 5 October 2015

Abstract

We describe a fluctuating volume-current formulation of electromagnetic fluctuations that extends our recent work on heat exchange and Casimir interactions between arbitrarily shaped homogeneous bodies [A. W. Rodriguez, M. T. H. Reid, and S. G. Johnson, Phys. Rev. B 88, 054305 (2013)] to situations involving incandescence and luminescence problems, including thermal radiation, heat transfer, Casimir forces, spontaneous emission, fluorescence, and Raman scattering, in inhomogeneous media. Unlike previous scattering formulations based on field and/or surface unknowns, our work exploits powerful techniques from the volume-integral equation (VIE) method, in which electromagnetic scattering is described in terms of volumetric, current unknowns throughout the bodies. The resulting trace formulas (boxed equations) involve products of well-studied VIE matrices and describe power and momentum transfer between objects with spatially varying material properties and fluctuation characteristics. We demonstrate that thanks to the low-rank properties of the associated matrices, these formulas are susceptible to fast-trace computations based on iterative methods, making practical calculations tractable. We apply our techniques to study thermal radiation, heat transfer, and fluorescence in complicated geometries, checking our method against established techniques best suited for homogeneous bodies as well as applying it to obtain predictions of radiation from complex bodies with spatially varying permittivities and/or temperature profiles.

Received 20 May 2015
Revised 12 August 2015

DOI:https://doi.org/10.1103/PhysRevB.92.134202

Authors & Affiliations

Athanasios G. Polimeridis¹, M. T. H. Reid², Weiliang Jin³, Steven G. Johnson², Jacob K. White⁴, and Alejandro W. Rodriguez³

¹Skolkovo Institute of Science and Technology, Moscow, Russia
²Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
³Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
⁴Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 92, Iss. 13 — 1 October 2015

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Schematic of a many-body geometry in which fluctuating current sources give rise to radiation as well as flux and momentum transfer between the bodies. Also illustrated are the incident field $ϕ_{inc}$ due to a single dipole source $σ$ within a body $V_{1}$ along with the induced polarization currents $ξ$ throughout $V_{1}$ and two nearby bodies, $V_{2}$ and $V_{3}$ , resulting in scattered fields $ϕ_{scat}$ . The characteristics of the dipole sources $σ$ (fluctuation statistics) and the permittivities of the bodies $χ$ (material properties) both vary within each object.
Reuse & Permissions
Figure 2
Flux spectrum $Φ (ω)$ of a cube of edge length $2 R$ held at temperature $T$ , normalized by the corresponding black-body spectrum $Φ_{BB} (ω) = \frac{A}{4 π^{2}} {(ω / c)}^{2} Θ (ω, T)$ , for different (a) discretization mesh densities and (b) rSVD truncation tolerances.
Reuse & Permissions
Figure 3
Flux spectrum $Φ (ω)$ normalized by the corresponding black-body spectrum $Φ_{BB} (ω) = \frac{A}{4 π^{2}} {(ω / c)}^{2} Θ (ω, T)$ of different bodies of surface area $A$ held at temperature $T = 1000$ K, including a sphere of radius $R$ (blue lines), cube of edge length $2 R$ (green line), and ellipsoid of long semiaxis $R$ and short semi axis $\frac{R}{2}$ (red line). The objects have either (a) uniform permittivities $ε = 12 + i$ or (b) spatially varying $ε (z) = ε_{- R} + (ε_{R} - ε_{- R}) \frac{| z + R |}{2 R}$ , with $ε_{R} = 12 + i$ and $ε_{- R} = 2 + i$ . For comparison, we also plot the radiation spectrum $Φ_{eff}$ (dashed lines) of corresponding bodies with homogeneous effective permittivities $ε_{eff} = 7 + i$ . The insets depict the angular distribution of far-field radiation $U (Ω)$ , normalized by the maximum intensity over all directions ${max}_{Ω} U$ , at selected frequencies.
Reuse & Permissions
Figure 4
Flux spectrum $Φ (ω)$ of various bodies normalized by the corresponding predictions of a simple approximation $Φ_{eff}$ , defined in Eq. (32), including (a) sphere of radius $R$ and radially varying temperature profile $T (r) = T_{0} + (T_{R} - T_{0}) \frac{r}{R}$ for both $T_{0} = 0, T_{R} = 1000$ K (blue line) and $T_{0} = 1000, T_{R} = 0$ (green line), and (b) sphere of radius $R$ (blue line) or ellipsoids with short semiaxes $\frac{R}{2}$ and long semiaxis $R$ along the $\hat{z}$ (green line) or $\hat{x}$ (red line) directions, subject to vertically varying temperature profiles $T (z) = T_{- L} + (T_{L} - T_{- L}) \frac{| z + L / 2 |}{L}$ , where $L$ denotes the $z$ dimension of the corresponding body. In all cases, objects have uniform permittivity $ε = 12 + i$ and are subject to temperature gradients $T_{- L} = 0$ and $T_{L} = 1000$ K. The insets in (a) show the cross-sections of the spatially varying flux contribution from dipoles in the sphere at different frequencies while those in (b) show the angular distribution of far-field radiation $U (Ω)$ normalized by ${max}_{Ω} U$ .
Reuse & Permissions
Figure 5
Heat-transfer spectrum $Φ (ω)$ , normalized by the corresponding black-body spectrum $Φ_{BB} (ω) = \frac{A}{4 π^{2}} {(ω / c)}^{2} Θ (ω, T)$ , between two cubes of edge length $2 R$ and temperature $T = 1000$ K separated by surface-surface distance $d = R$ . The cubes are assumed to have either uniform permittivities $ε = 2 + i$ (red dashed line), $ε = ε_{eff} = 7 + i$ (black dashed line), or $ε = 12 + i$ (blue dashed line), or vertically varying permittivities $ε (z_{i}) = ε_{- R} + (ε_{R} - ε_{- R}) \frac{| z_{i} + R |}{2 R}$ defined with respect to the local axis $x_{1, 2}$ at the center of each cube (shown on the inset), chosen so that the system has mirror symmetry about the $x - y$ plane intersecting the origin $O$ . The gradients are either increasing (black solid line) or decreasing (green solid line) toward or away from the center, corresponding to the choice of $ε_{R, - R} = {12 + i, 2 + i}$ or $ε_{R} \leftrightarrow ε_{- R}$ , respectively.
Reuse & Permissions
Figure 6
Far-field fluorescence spectrum $Φ (ω)$ (in arbitrary units) of a homogeneous and nondispersive dielectric sphere of radius $R$ and permittivity $ε = 12 + i$ excited by an $\hat{x}$ -polarized planewave propagating along the $\hat{z}$ direction with frequency $ω_{inc} R / c = 1.58$ . The absorbed power $χ_{inc} (x)$ inside the sphere, obtained by solving a single scattering problem as described in Ref. [25], is shown in the top contour plots along three sphere cross-sections. $Φ$ is computed exactly (blue line) or via a homogeneous approximation $Φ_{eff}$ in which the absorbed power is taken to be uniformly distributed inside the sphere and given by $χ_{eff} = \int_{V} d^{3} x χ_{inc} (x)$ (red line). The ratio of the two is plotted as the black dashed line on the right axis. The insets depict the angular distribution of fluorescence emission, normalized by the maximum intensity over all directions, at selected frequencies.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics