In this case,[1] spectral flux density is the quantity that describes the rate at which energy transferred by electromagnetic radiation is received from that unresolved point source, per unit receiving area facing the source, per unit wavelength range.
At any given wavelength λ, the spectral flux density, Fλ, can be determined by the following procedure: Spectral flux density is often used as the quantity on the y-axis of a graph representing the spectrum of a light-source, such as a star.
There are two main approaches to definition of the spectral flux density at a measuring point in an electromagnetic radiative field.
The vector definition seems to be preferred for theoretical investigations of the physics of the radiative field.
In this case, nature tells the investigator what is the magnitude, direction, and sense of the flux density at the prescribed point.
Mathematically, defined as an unweighted integral over the solid angle of a full sphere, the flux density is the first moment of the spectral radiance (or specific intensity) with respect to solid angle.
[5] It is not common practice to make the full spherical range of measurements of the spectral radiance (or specific intensity) at the point of interest, as is needed for the mathematical spherical integration specified in the strict definition; the concept is nevertheless used in theoretical analysis of radiative transfer.
This investigator thinks of a unit area in a horizontal plane, surrounding the prescribed point.
The investigator wants to know the total power of all the radiation from the atmosphere above in every direction, propagating with a downward sense, received by that unit area.
[10][11][12][13][14] For the flux density scalar for the prescribed direction and sense, we may write where with the notation above,
is not a vector because it is a positive scalar-valued function of the prescribed direction and sense, in this example, of the downward vertical.
In a region in which the material is uniform and the radiative field is isotropic and homogeneous, let the spectral radiance (or specific intensity) be denoted by I (x, t ; r1, ν), a scalar-valued function of its arguments x, t, r1, and ν, where r1 denotes a unit vector with the direction and sense of the geometrical vector r from the source point P1 to the detection point P2, where x denotes the coordinates of P1, at time t and wave frequency ν.
In this case, the value of the vector flux density at P1 is the zero vector, while the scalar or hemispheric flux density at P1 in every direction in both senses takes the constant scalar value πI.
The scalar or hemispheric spectral flux density is convenient for discussions in terms of the two-stream model of the radiative field, which is reasonable for a field that is uniformly stratified in flat layers, when the base of the hemisphere is chosen to be parallel to the layers, and one or other sense (up or down) is specified.
In an inhomogeneous non-isotropic radiative field, the spectral flux density defined as a scalar-valued function of direction and sense contains much more directional information than does the spectral flux density defined as a vector, but the full radiometric information is customarily stated as the spectral radiance (or specific intensity).
[17] The spectral radiance (or specific intensity) is suitable for the description of an uncollimated radiative field.
The integrals of spectral radiance (or specific intensity) with respect to solid angle, used above, are singular for exactly collimated beams, or may be viewed as Dirac delta functions.
Therefore, the specific radiative intensity is unsuitable for the description of a collimated beam, while spectral flux density is suitable for that purpose.
[18] At a point within a collimated beam, the spectral flux density vector has a value equal to the Poynting vector,[8] a quantity defined in the classical Maxwell theory of electromagnetic radiation.
[7][19][20] Sometimes it is more convenient to display graphical spectra with vertical axes that show the relative spectral flux density.
In this case, the spectral flux density at a given wavelength is expressed as a fraction of some arbitrarily chosen reference value.
Relative spectral flux densities are expressed as pure numbers without any units.