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On the Wave Speed of Thermal Radiation Inside and Near the Boundary of an Absorbing Material

机译：吸收材料边界内外的热辐射的波速

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摘要

Planck's law describes thermal radiation into vacuum from a black body in thermal equilibrium. This law can be easily adapted to describe radiation into a transparent medium with a constant refractive index, and it admits a less trivial extension to radiation into a transparent medium with a nonconstant refractive index. However, this law cannot be straightforwardly generalized to describe thermal radiation into absorbing media and, in particular, to describe thermally exited electromagnetic fields inside the radiating body itself. We first analyze Planck's law and show why it cannot be straightforwardly extended to radiation into an absorbing medium. The derivation of this law relies on the assumption that a radiated field admits decomposition into normal modes, which cannot exist in absorbing media that are characterized by a complex-valued refractive index n = n' + in", whose imaginary part describes the rate of energy dissipation. Correspondingly, the speed of electromagnetic waves in absorbing media c = c_0, where c_0 is the speed of light in vacuum, is also complex-valued, which suggests that the conventional concept of a complex valued wave speed is not suitable for modeling thermal radiation. We demonstrate that complex-valued wave speeds adequately describe waves that carry signals, such as radio waves and laser beams. Such waves decay because they pass some of their energy to the medium. The energy absorbed by the medium is eventually reradi-ated, but in studies focused on the transmission of signals, the reradiated fields are ignored as noise. In order to study thermal radiation in an absorbing material, one must treat the material and the radiation together as a closed system. The energy in such a system is conserved, and its distribution between the material and radiation does not change in time. This radiation admits decomposition into normal modes, which makes it possible to extend Planck's law to radiation into absorbing materials. This paper proposes a model of thermal radiation in an absorbing medium as a closed, energy conserving system. The radiation field has normal modes that correspond to an effective speed of wave propagation. Assuming that an absorbing material and the radiation in it are in thermal equilibrium, we show that deep inside the material, the average speed of photons is given by a frequency and temperature-dependent expression c_* = c_0/(1 + e~(-hω/κT)). While this result is independent of the material, we further show that close to the boundary of the medium, the speed of thermal radiation depends in a complex way on the refractive index and the extinction coefficient of the material, as well as the direction of propagation and the distance from the material's surface.
机译：普朗克定律描述了处于热平衡状态的黑体向真空的热辐射。该定律可以很容易地适用于描述向具有恒定折射率的透明介质中的辐射，并且它允许向具有非恒定折射率的透明介质中的辐射的平凡程度较小。但是，该定律不能简单地概括为将热辐射描述为吸收介质，特别是描述辐射体本身内部的热辐射电磁场。我们首先分析普朗克定律，并说明为什么不能直接将其扩展到吸收介质中的辐射。该定律的推导基于这样一个假设，即辐射场允许分解为正常模式，而在以复数值n = n'+ in“为复数值的吸收介质中不存在该模式，其虚部描述了速率。相应地，电磁波在吸收介质中的速度c = c_0 / n，其中c_0是真空中的光速，也是复数值，这表明复数值波速的传统概念不适合我们证明了复数值波速足以描述携带信号的波，例如无线电波和激光束，这些波会衰减，因为它们将部分能量传递给介质，最终被介质吸收的能量辐射，但在专注于信号传输的研究中，将辐射场视为噪声，因此，为了研究吸收性材料中的热辐射，必须处理材料和辐射一起作为一个封闭的系统。这样的系统中的能量是守恒的，并且它在材料和辐射之间的分布不会随时间变化。这种辐射允许分解为正常模式，从而有可能将普朗克定律扩展到辐射到吸收材料中。本文提出了一种吸收介质中的热辐射模型，作为一个封闭的节能系统。辐射场具有与波的有效传播速度相对应的正常模式。假设吸收材料和其中的辐射处于热平衡状态，则表明在材料内部，光子的平均速度由频率和温度相关的表达式c_ * = c_0 /（1 + e〜（- hω/κT））。尽管此结果与材料无关，但我们进一步表明，靠近介质边界，热辐射的速度以复杂的方式取决于材料的折射率和消光系数以及传播方向以及与材料表面的距离。

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  • 来源
    《Journal of Heat Transfer》 |2020年第3期|032801.1-032801.8|共8页
  • 作者

    Bair V. Budaev; David B. Bogy;

  • 作者单位

    Department of Mechanical Engineering University of California Berkeley CA 94720;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);
  • 原文格式 PDF
  • 正文语种 eng
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