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Spatial Coherence of Thermal Emission by a Sphere

Analytical expressions for calculating the energy density and spatial correlation function of thermal emission by a homogeneous, isothermal sphere of arbitrary size and material are presented. The spectral distribution and the power law governing the distance-dependent energy density are investigated in the near-field and far-field regimes for silicon carbide, silicon and tungsten spheres of various size parameters ranging from X = 0.002 to 5. The spatial coherence of thermal field emitted by spheres is also studied in both radial and polar directions. The energy density follows a power law of d^-2 (d is the observation distance) in the far field for all sizes and materials. The power law in the near field is strongly dependent on the material, size parameter, and the ratio d/a (a is the sphere radius). In the near field, the energy density follows a power law of d^-6 when X<<1 and d/a>>1 (similar to an electric point dipole). With increasing X or decreasing d/a, the contribution of multipoles to the energy density increases resulting in an increase in the power of d until the power law converges to that for a semi-infinite medium. The spatial correlation length in the radial direction is in the orders of $λ$, 0.1$λ$, and 0.001$λ$ in the far field, intermediate near field, and extreme near field, respectively. The correlation angle in the extreme near field is strongly dependent on the sphere size parameter, such that it decreases by three orders of magnitude (from 0.5$π$ to 0.001$π$) when X increases from 0.002 to 5. In the intermediate near field and far field, the correlation angle retains the same order of magnitude (0.15$π$ - 0.7$π$) for all considered Xs. While the excitation of dipolar localized surface phonons (LSPhs) does not affect the correlation length and angle, the multipolar LSPhs reduce the spatial coherence in both directions.