The Poynting vector, specific intensity, and net flux

The Planck function can also be expressed in terms of wavelength (Av = 1 X dv + 

The Planck function appears in many aspects of the theory of radiative transfer and the design of infrared instmmentation, as is discussed in forthcoming chapters. 

The energy flux of a plane, monochromatic wave is represented by the Poynting vector as discussed in Section 1.2. In this section we relate the Poynting vector to other quantities used in the description of the radiative transfer of energy in planetary atmospheres and from surfaces. 

Strictly monochromatic radiation propagating in a unique direction (e.g., from a point source) is never realized. A monochromatic wave implies a periodic process of infinite duration. Such waves do not exist, although the signal from a stable, singlemode laser provides a fair approximation. Ordinary incoherent radiation emitted and reflected from real atmospheres and surfaces consists of individual wave packets of finite length and duration a few meters and 10 seconds are typical values. Similarly, point sources are replaced by extended sources in practice. Radiation from such sources tends to be incoherent and covers a range of frequencies and directions. Thus, it is more convenient to work with a distribution of plane waves and their associated Poynting vectors. 

Consider radiation incident on an element of area da with unit normal vector n as shown in Fig. 1.8.1. The radiation can be regarded as an incoherent superposition of plane waves, each with an associated Poynting vector S(v, k) where v is the wavenumber of the wave and k is a unit vector defining the direction of propagation. Let N S, V, k) d5 dv dco be the number of plane waves with Poynting vector 

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