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Convergence of vector spherical wave expansion method applied to near-field radiative transfer

Karthik Sasihithlu and Arvind Narayanaswamy

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Abstract

Near-field radiative transfer between two objects can be computed using Rytov’s theory of fluctuational electrodynamics in which the strength of electromagnetic sources is related to temperature through the fluctuation-dissipation theorem, and the resultant energy transfer is described using the dyadic Green’s function of the vector Helmholtz equation. When the two objects are spheres, the dyadic Green’s function can be expanded in a series of vector spherical waves. Based on comparison with the convergence criterion for the case of radiative transfer between two parallel surfaces, we derive a relation for the number of vector spherical waves required for convergence in the case of radiative transfer between two spheres. We show that when electromagnetic surface waves are active at a frequency the number of vector spherical waves required for convergence is proportional to Rmax/d when d/Rmax → 0, where Rmax is the radius of the larger sphere, and d is the smallest gap between the two spheres. This criterion for convergence applies equally well to other near-field electromagnetic scattering problems.

© 2011 Optical Society of America

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Figures (6)
Fig. 1
Fig. 1 (a) The configuration of two spheres for which the study is performed. (b) The variation of real and imaginary part of the dielectric function (ε) of silica in the frequency range under consideration.

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Fig. 2
Fig. 2 Convergence of conductance (on the left axis) and error (on the right axis) shown for (a) R = 10 μm spheres at d = 100 nm and (b) R = 25 μm spheres at d = 250 nm. The solid line through the relative error data points is included to illustrate the exponentially decaying trend.

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Fig. 3
Fig. 3 Convergence of spectral conductance (on the left axis) and error (on the right axis) shown for R = 10 μm spheres for d = 100 nm at (a) a nonresonant frequency (0.1005 eV) and (b) a resonant frequency (0.061 eV).

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Fig. 4
Fig. 4 Variation of Nconv with R/d for two equal-sized spheres with R = 500 nm, 1 μm, 15 μm and 25 μm.

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Fig. 5
Fig. 5 Convergence of spectral conductance shown for R 1 = 2 μm and R 2 = 40 μm spheres for d = 200 nm at (a) a nonresonant frequency (0.1005 eV) (b) a resonant frequency (0.061 eV).

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Fig. 6
Fig. 6 Contribution to spectral conductance from each value of m for R = 25 μm and d = 250 nm at (a) a resonant frequency (0.061 eV) (b) nonresonant frequency (0.1005 eV). The rate of exponential decay (B) for higher values of m at the resonant frequency is also shown.

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Equations (14)

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∇   ×   ∇   ×   A ( r )   −   k 2 A ( r )   = 0 ,
M lm ( p )   ( k r )   =   z l ( p )   ( kr ) V lm ( 2 )   ( θ ,   ϕ )
N lm ( p )   ( k r )   =   ζ l ( p )   ( kr ) V lm ( 3 )   ( θ ,   ϕ )   +   z l ( p )   ( kr ) kr     l ( l   +   1 ) V lm ( 1 )   ( θ ,   ϕ )
E   ( r )   =   ∑ l = 1 ∞ ∑ m = − l l ( A lm ( p ) M lm ( p )   ( k r )   +   B lm ( p ) N lm ( p )   ( k r ) )
H   ( r )   =   ik ω μ   ∑ l = 1 ∞ ∑ m = − l l ( A lm ( p )   N lm ( p )   ( k r )   +   B lm ( p )   M lm ( p )   ( k r ) )
N conv   =   a   +   x   +   b x 1 / 3
N conv   =   e π D λ ,
E   ( r )   =   ∑ m = 0 ∞ ∑ l = ( m , 1 ) ∞ ( A lm ( p )   M lm ( p )   ( k r )   +   B lm ( p )   N lm ( p )   ( k r ) )
H   ( r )   =   ik ω μ   ∑ m = 0 ∞ ∑ l = ( m , 1 ) ∞ ( A lm ( p ) N lm ( p )   ( k r )   +   B lm ( p )   M lm ( p )   ( k r ) ) ,
G   =   lim T 1 → T 2   Q ( T 1 ,   T 2 ) T 1   −   T 2 ,
N conv   =   C R d ,
N conv   =   C R d   +   e π D λ ,
N conv   =   C R 2 d   +   e π D λ ,
G ω ( M conv )   =   0.005 G ω ( 0 )
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