The determination of emissivity of layered structures is critical in many applications, such as radiation thermometry, microelectronics, radiative cooling, and energy harvesting. Two different approaches, i.e., the “indirect” and “direct” methods, are commonly used for computing the emissivity of an object. For an opaque surface at a uniform temperature, the indirect method involves calculating the spectral directional-hemispherical reflectance to deduce the spectral directional emissivity based on Kirchhoff’s law. On the other hand, a few studies have used a combination of Maxwell’s equations with the fluctuation-dissipation theorem to directly calculate the emissivity. The present study aims at unifying the direct and indirect methods for calculating the far-field thermal emission from layered structures with a nonuniform temperature distribution. Formulations for both methods are given to illustrate the equivalence between the indirect and the direct methods. Thermal emission from an asymmetric Fabry–Pérot resonance cavity with a nonuniform temperature distribution is taken as an example to show how to predict the intensity, emissivity, and the brightness temperature. The local density of states, however, can only be calculated using the direct method.

The unpolished surface of crystalline silicon wafers often exhibits nonGaussian and anisotropic roughness characteristics, as evidenced by the side peaks in the slope distribution. This work investigates the effect of anisotropy on the emittance. The directional-hemispherical reflectance of slightly and strongly anisotropic silicon wafers was measured at room temperature using a center-mount integrating sphere. A monochromator with a lamp was used for near-normal incidence in the wavelength region from 400-1000 nm, and a continuous-wave diode laser at the wavelength of 635 nm was used for measurements at zenith angles up to 60 • . The directional emittance was deduced from the measured reflectance based on Kirchhoff's law. The geometricoptics-based Monte Carlo model that incorporates the measured surface topography is in good agreement with the experiment. Both the experimental and modeling results suggest that anisotropic roughness increases multiple scattering, thereby enhancing the emittance. On the other hand, if the wafer with strongly anisotropic roughness were modeled as a Gaussian surface with the same roughness parameters, the predicted emittance near the normal direction would be lower by approximately 0.05, or up to 10% at a wavelength of 400 nm. Comparisons also suggest that the Gaussian surface assumption is questionable in calculating the emittance at large emission angles with s polarization, even for the slightly anisotropic wafer. This work demonstrates that anisotropy plays a significant role in the emittance enhancement of rough surfaces. Hence, it is imperative to obtain precise surface microstructure information in order to accurately predict the emittance, a critical parameter for non-contact temperature measurements and radiative transfer analysis. 9180195-928X/07/0600-0918/0

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An understanding of the variation of directional radiative properties of rough surfaces with dielectric coatings is important for temperature measurements and heat transfer analysis in many industrial processes. An experimental study has been conducted to investigate the effect of coating thickness on the bidirectional reflectance distribution function (BRDF) of rough silicon surfaces. Silicon dioxide films with thicknesses of 107.2, 216.5, and 324.6 nm were deposited using plasma-enhanced chemical vapor deposition onto the rough side of two Si wafers. The wafer surfaces exhibit distinct anisotropic characteristics as a result of chemical etching during the manufacturing process. A laser scatterometer measures the BRDF at a wavelength of 635 nm, after improvement of the signal-to-noise ratio. The slope distribution function obtained from the measured BRDF of uncoated Si surfaces was used in an analytical model based on geometric optics for rough surface scattering and thin-film optics for microfacet reflectance. The predicted BRDFs are in reasonable agreement with experimental results for a large range of coating thicknesses. The limitations of the geometric optics for modeling the BRDF of coated anisotropic rough surfaces in the specular direction are demonstrated. The results may benefit future radiative transfer analysis involving complicated surface microstructures with thin-film coatings.

The determination of emissivity of layered structures is critical in many applications, such as radiation thermometry, microelectronics, radiative cooling, and energy harvesting. Two different approaches, i.e., the “indirect” and “direct” methods, are commonly used for computing the emissivity of an object. For an opaque surface at a uniform temperature, the indirect method involves calculating the spectral directional-hemispherical reflectance to deduce the spectral directional emissivity based on Kirchhoff’s law. On the other hand, a few studies have used a combination of Maxwell’s equations with the fluctuation-dissipation theorem to directly calculate the emissivity. The present study aims at unifying the direct and indirect methods for calculating thermal emission from layered structures with a nonuniform temperature distribution. Formulations for both methods are given to illustrate the equivalence between the indirect and the direct methods. Thermal emission from an asymmetric Fabry-Perot resonance cavity with a nonuniform temperature distribution is taken as an example to show how to predict the intensity, emissivity, and the brightness temperature. The local density of states, however, can only be calculated using the direct method.