Definition of radiometric quantities

Radiant energy and radiant flux. Radiation carries energy that can be absorbed in matter heating up the absorber or interacting with electrical charges.Radiant energyQis measured in units of Joule (1 J = 1 Ws). It quantifies the total energy emitted by a source or received by a detector.

Radiant flux Φis defined as radiant energy per unit time interval Φ= dQ

dt (2.6)

passing through or emitted from a surface. Radiant flux has the unit watts (W) and is also frequently called radiant power, which corre-sponds to its physical unit. Quantities describing the spatial and ge-ometric distributions of radiative flux are introduced in the following sections.

The units for radiative energy, radiative flux, and all derived quan-tities listed in Table2.1 are based on Joule as the fundamental unit.

Instead of theseenergy-derivedquantities an analogous set of photon-derived quantities can be defined based on the number of photons.

Photon-derived quantities are denoted by the subscript p, while the energy-based quantities are written with a subscripteif necessary to distinguish between them. Without a subscript, all radiometric quanti-ties are considered energy-derived. Given the radiant energy the num-ber of photons can be computed from Eq. (2.2)

Np= Qe

ep = λ

hcQe (2.7)

With photon-based quantities the number of photons replaces the ra-diative energy. The set of photon-related quantities is useful if radia-tion is measured by detectors that correspond linearly to the number of absorbed photons (photon detectors) rather than to thermal energy stored in the detector material (thermal detector).

Photon fluxΦp is defined as the number of photons per unit time interval

Φp= dNp

dt ⁼ λ hc

dQe

dt ⁼ λ

hcΦe (2.8)

Similarly, all other photon-related quantities can be computed from the corresponding energy-based quantities by dividing them by the energy of a single photon.

Because of the conversion from energy-derived to photon-derived quantities Eq. (2.7) depends on the wavelength of radiation. Spectral distributions of radiometric quantities will have different shapes for both sets of units.

2.3 Radiometric quantities 21 a

dS

b

dS

Figure 2.5:Illustration of the radiometric quantities:aradiant exitance; andb irradiance. (By C. Garbe, University of Heidelberg.)

Radiant exitance and irradiance. Radiant exitanceM defines the ra-diative fluxemittedper unit surface area

M= dΦ

dS (2.9)

of a specified surface. The flux leaving the surface is radiated into the whole hemisphere enclosing the surface elementdSand has to be inte-grated over all angles to obtainM (Fig.2.5a). The flux is, however, not radiated uniformly in angle. Radiant exitance is a function of position on the emitting surface,M=M(x). Specification of the position on the surface can be omitted if the emitted fluxΦis equally distributed over an extended areaS. In this caseM=Φ/S.

IrradianceEsimilarly defines the radiative fluxincidenton a certain point of a surface per unit surface element

E= dΦ

dS (2.10)

Again, incident radiation is integrated over all angles of the enclosing hemisphere (Fig.2.5b). Radiant exitance characterizes an actively radi-ating source while irradiance characterizes a passive receiver surface.

Both are measured in W m⁻²and cannot be distinguished by their units if not further specified.

Radiant intensity. Radiant intensityIdescribes the angular distribu-tion of radiadistribu-tion emerging from a point in space. It is defined as radiant flux per unit solid angle

I= dΦ

dΩ (2.11)

and measured in units of W sr⁻¹. Radiant intensity is a function of the direction of the beam of radiation, defined by the spherical coordinates

a

Figure 2.6: Illustration of radiometric quantities: aradiant intensity; and b radiance. (By C. Garbe, University of Heidelberg.)

θandφ(Fig.2.6). Intensity is usually used to specify radiation emitted frompoint sources, such as stars or sources that are much smaller than their distance from the detector, that is,dxdyr². In order to use it for extended sources those sources have to be made up of an infinite number of infinitesimal areas. The radiant intensity in a given direc-tion is the sum of the radiant flux contained in all rays emitted in that direction under a given solid angle by the entire source (see Eq. (2.18)).

The term intensity is frequently confused with irradiance or illumi-nance. It is, however, a precisely defined quantity in radiometric termi-nology and should only be used in this context to avoid confusion.

Radiance. RadianceLdefines the amount of radiant flux per unit solid angle per unit projected area of the emitting source

L= d²Φ

dΩdS⊥ = d²Φ

dΩdScosθ (2.12)

where dS⊥= dScosθ defines a surface element that is perpendicular to the direction of the radiated beam (Fig.2.6b). The unit of radiance is W m⁻²sr⁻¹. Radiance combines the concepts of exitance and intensity, relating intensity in a certain direction to the area of the emitting sur-face. And conversely, it can be thought of as exitance of the projected area per unit solid angle.

Radiance is used to characterize an extended source that has an area comparable to the squared viewing distance. As radiance is a function of both position on the radiating surface as well as direction L=L(x,θ,φ), it is important always to specify the point in the surface and the emitting angles. It is the most versatile quantity in radiometry as all other radiometric quantities can be derived from the radiance integrating over solid angles or surface areas (Section2.3.4).

2.3 Radiometric quantities 23

Figure 2.7:Illustration of spherical coordinates.