We define the energy and momentum of a photon as follows, where h is Planck's constant, c is the speed of light, is the frequency of the light, and s is a unit vector which points in the direction of propagation.

We begin by considering the macroscopic properties of the radiation field. In the absence of diffraction effects, radiation can be treated as traveling in straight lines (rays). We consider the amount of energy dE which crosses through an area s . dA, where is the angle between the normal to the surface area and s, into a solid angle d_s in time dt with frequency range d.

[NMSU, N. Vogt]

where

and the specific intensity I has units of erg/s/cm²/str/Hz. Although I is expressed as a scalar quantity, it is defined as the intensity in a particular direction (along the normal ray in the above diagram). The exception to this rule is the case of an isotropic radiation field, for which the specific intensity is the same in all directions. For a blackbody,

The solid angle d_s is defined so that, integrated over a sphere, it is equal to 4.

The mean intensity J is the zeroth moment of the radiation field. This is the average of I over all solid angles. For isotropic radiation, J = I.

The energy density u is the amount of radiant energy per unit volume at frequency . Visualize a cylinder of side area dA and length c × dt, drawn around a set of light rays traveling through d_s. There is a certain amount of energy within the cylinder. As the speed of propagation of light (in a vacuum) is c,

The net flux density F in the direction z at frequency is the integral of I cos over all solid angles. This is equivalent to the net energy flow through the area dA into the solid angle d_s. For an isotropic radiation field, the net energy flow is zero.

We define H as the first moment of the intensity I.

The second moment K is then

which is related to the radiation pressure p_r as follows. The momentum flux, integrated over all frequencies along the direction s is equal to dF/c. The radiation pressure is simply the component of the momentum flux in the z direction.