When the features of a surface or the size of particles is of the same order as the wavelength of the incident radiation, then some effects differ from those discussed for traditional engineering applications of radiative heat transfer. Radiation can generally be treated as an electromagnetic wave until the wavelength becomes of the order of the size of the atoms in a solid interacting with the radiation. Thus, gamma rays have effective wavelengths that are of the order of atomic sizes, and gamma rays do not undergo surface refraction as governed by Maxwell's equations. Radiation in the range of wavelengths common in engineering heat transfer can generally be treated by electromagnetic theory.

The propagation of thermal radiation can be treated in terms of electromagnetic waves. Maxwell's equations relating the electric field vector E and the magnetic field vector H are

$\nabla X H = \gamma (\delta E/\delta t) + (E\ {r_e} )$
$\nabla X E = -\mu (\delta H/\delta t)$
$\nabla \dot E = 0$

$\nabla X H = 0$

For plane waves propagating in an unbounded medium, the form of the electric field Ey (Fig. 1) from Basics of Radiation that satisfies Maxwell's equations is (Bohren and Huffman, 1983)

${E_y} = {E_{y,o}}\exp \left\{ {i\omega \left[ {t - (n - i\kappa )\frac{x}{{{c_o}}}} \right]} \right\} = {E_{y,o}}\exp ( - \frac{{2\pi \kappa x}}{\lambda })\exp (i\omega t - i\frac{{2\pi nx}}{\lambda })\qquad \qquad(2)$

where Eo is the maximum wave amplitude, ω is the angular frequency ω = 2πco / λ, n is the simple refractive index, κ is the imaginary component of the complex refractive index, $\bar n = n - i\kappa$, and λ is the wavelength in a vacuum.

The magnitude of the energy carried by this plane wave $\left| {\mathbf{S}} \right|$, is

$\left| {\mathbf{S}} \right| = \frac{{\bar n}}{{\mu {c_o}}}E_y^2\qquad \qquad(3)$

where μ is the magnetic permeability of the medium and S is the Poynting vector, E×H. The value of μ in a vacuum, μo, is 4πx10-7N/A2. The radiative intensity I is directly proportional to $\left| {\mathbf{S}} \right|$.

The interaction of a plane wave propagating through one medium and interacting at the smooth interface (relative to the wavelength) with a second material with differing properties can be analyzed using Equations (1)-(3), and is the basis for the property predictions.

When a plane wave traveling through space interacts with a spherical dielectric particle of differing refractive index, part of the wave is refracted into the particle and the complex patterns of interference and reflection within the particle give a highly lobed structure for the radiation reflected from the particle. The exact form of the radiated pattern depends on the complex refractive index as well as the ratio of particle diameter to wavelength. Gustav Mie (1908), is usually given credit for the analytical solution for this case. The complete analytical solution includes terms that decay rapidly with distance from the particle (within a very few wavelengths), and these decaying fields are called evanescent waves. These near-field terms are usually neglected in the so-called Mie scattering solutions, since the particles in most suspensions (interstellar dust, ash particles in combustion products, atmospheric aerosols, etc.) are quite far apart on a wavelength scale. Neglect of the terms is unimportant in these cases, and the resulting solutions are called far-field solutions. However, for particles that are close to another object or surface, the evanescent waves greatly change the energy transfer between the particle and object, and their effect must be included. Heltzel, 2006 studied the effect of interaction of a continuous-wave laser with a spherical silica sphere. Very near the sphere (in the near-field), the laser energy is greatly enhanced in the forward direction, resulting in a 15-fold increase in the peak forward intensity just past the sphere itself. This effect of magnifying the laser intensity over a very small area can be used as a means to focus intense laser energy onto a substrate at nanometer scales.

## References

Bohren, C.F. and Huffman D.R., 1983, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York, NY.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Heltzel, A.J., 2006, Laser/Microstructure Interaction and Ultrafast Heat Transfer, PhD Dissertation, Mechanical Engineering Department, The University of Texas at Austin, Austin, TX.

Mie, G., 1908, “Beiträge zur Optik trüber Medien speziell kolloidaller Metallösungen,” Ann. Phys., Vol. 25, pp. 377-445.