r
φ
s
Figure 2.3:Definition of plane angle.
2.3 Radiometric quantities
2.3.1 Solid angleIn order to quantify the geometric spreading of radiation leaving a source, it is useful to recall the definition of solid angle. It extends the concept of plane angle into 3-D space. Aplane angle θis defined as the ratio of the arc lengths on a circle to the radiusr centered at the point of definition:
θ= s
r (2.3)
The arc lengths can be considered as projection of an arbitrary line in the plane onto the circle (Fig.2.3). Plane angles are measured in rad (radians). A plane angleθquantifies the angular subtense of a line segment in the plane viewed from the point of definition. A circle has a circumference of 2πr and, therefore, subtends a plane angle of 2πrad. Asolid angleΩis similarly defined as the ratio of an areaAon the surface of a sphere to the square radius, as shown in Fig.2.4:
Ω=rA2 (2.4)
The area segmentAcan be considered as the projection of an arbitrarily shaped area in 3-D space onto the surface of a sphere. Solid angles are measured in sr (steradian). They quantify the areal subtense of a 2-D surface area in 3-D space viewed from the point of definition. A sphere subtends a surface area of 4πr2, which corresponds to a solid angle of 4πsr. Given a surface areaAthat is tilted under some angleθbetween the surface normal and the line of sight the solid angle is reduced by a factor of cosθ:
Ω= A
r W
x y
z
A
Figure 2.4: Definition of solid angle. (By C. Garbe, University of Heidelberg.)
Table 2.1: Definitions of radiometric quantities (corresponding photometric quantities are defined in Table2.2)
Quantity Symbol Units Definition
Radiant energy Q Ws Total energy emitted by a source
or received by a detector
Radiant flux Φ W Total power emitted by a source
or received by a detector
Radiant exitance M W m−2 Power emitted per unit surface
area
Irradiance E W m−2 Power received at unit surface
element
Radiant intensity I W sr−1 Power leaving a point on a sur-
face into unit solid angle
Radiance L W m−2sr−1 Power leaving unit projected sur-
face area into unit solid angle
From the definition of angles as ratios of lengths or areas it follows that they have no physical unit. However, it is advisable always to use the artificial units rad and sr when referring to quantities related to angles to avoid confusion. Radiometric and photometric quantities also have to be defined carefully as their meaning cannot be inferred from physical units (Tables2.1and2.2).
2.3.2 Conventions and overview
Measurements of radiometric and photometric quantities very often are subject to confusion related to terminology and units. Due to di- verse historical developments and often inaccurate usage of names, radiometry is one of the least understood subjects in the field of op-
2.3 Radiometric quantities 19
Table 2.2: Definitions of photometric quantities (corresponding radiometric quantities are defined in Table2.1)
Quantity Symbol Units Definition
Luminous energy Qν lm s
Total luminous energy
emitted by a source or received by a detector
Luminous flux Φν lm (lumen)
Total luminous power
emitted by a source or received by a detector
Luminous exitance Mν lm m−2 Luminous power emitted
per unit surface area
Illuminance Eν lm m
−2 = lx (lux)
Luminous power received at unit surface element
Luminous intensity Iν lumen sr−1
= cd (candela)
Luminous power leaving a point on a surface into unit solid angle
Luminance Lν lumen m
−2sr−1 = cd m−2
Luminous power leaving
unit projected surface
area into unit solid angle
tics. However, it is not very difficult if some care is taken with regard to definitions of quantities related to angles and areas.
Despite confusion in the literature, there seems to be a trend to- wards standardization of units. (In pursuit of standardization we will use only SI units, in agreement with the International Commission on Illumination CIE. The CIE is the international authority defining termi- nology, standards, and basic concepts in radiometry and photometry. The radiometric and photometric terms and definitions are in com- pliance with the American National Standards Institute (ANSI) report RP-16, published in 1986. Further information on standards can be found at the web sites of CIE (http://www.cie.co.at/cie/) and ANSI (http://www.ansi.org), respectively.)
In this section, the fundamental quantities of radiometry will be defined. The transition to photometric quantities will be introduced by a generic Equation (2.27), which can be used to convert each of these radiometric quantities to its corresponding photometric counterpart.
We will start from the concept of radiative flux and derive the most important quantities necessary to define the geometric distribution of radiation emitted from or irradiated on surfaces. The six fundamen- tal concepts relating the spatial distribution of energy in electromag- netic radiation are summarized in Table2.1. The term “radiant” is only added to the names of those quantities that could be confused with the corresponding photometric quantity (see Table2.2).
2.3.3 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 ofphoton- 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= Qee p =
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 radiation 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 Z Y X W q f d b Z Y X W q f d dS dS = dS cos q
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 φ dφ dΩ= sinθ θ φd d dΩ dS θ dθ r
Figure 2.7:Illustration of spherical coordinates.
2.3.4 Relationship of radiometric quantities
Spatial distribution of exitance and irradiance. Solving Eq. (2.12)
for dΦ/dSyields the fraction of exitance radiated under the specified direction into the solid angledΩ
dM(x)= d dΦ dS =L(x,θ,φ)cosθdΩ (2.13) Given the radianceL of an emitting surface, the radiant exitance M can be derived by integrating over all solid angles of the hemispheric enclosureH: M(x)= H L(x,θ,φ)cosθdΩ= 2π 0 π/2 0 L(x,θ,φ)cosθsinθdθdφ (2.14) In order to carry out the angular integrationspherical coordinateshave been used (Fig.2.7), replacing the differential solid angle element dΩ by the two plane angle elements dθand dφ:
dΩ=sinθdθdφ (2.15) Correspondingly, the irradianceEof a surfaceScan be derived from a given radiance by integrating over all solid angles of incident radiation:
E(x)= H L(x,θ,φ)cosθdΩ= 2π 0 π/2 0 L(x,θ,φ)cosθsinθdθdφ (2.16)
Angular distribution of intensity. Solving Eq. (2.12) for dΦ/dΩyields
dS dI= d dΦ dΩ =L(x,θ,φ)cosθdS (2.17) Extending the point source concept of radiant intensity to extended sources, the intensity of a surface of finite area can be derived by inte- grating the radiance over the emitting surface areaS:
I(θ,φ)=
S
L(x,θ,φ)cosθdS (2.18)
The infinitesimal surface area dSis given by dS= ds1ds2, with thegen- eralized coordinatess =[s1, s2]T defining the position on the surface. For planar surfaces these coordinates can be replaced byCartesian co- ordinatesx=[x,y]T in the plane of the surface.
Total radiant flux. Solving Eq. (2.12) for d2Φ yields the fraction of
radiant flux emitted from an infinitesimal surface element dS under the specified direction into the solid angle dΩ
d2Φ=L(x,θ,φ)cosθdS dΩ (2.19) The total flux emitted from the entire surface areaSinto the hemispher- ical enclosureH can be derived by integrating over both the surface area and the solid angle of the hemisphere
Φ= S H L(x,θ,φ)cosθdΩdS= S 2π 0 π/2 0 L(x,θ,φ)cosθsinθdθdφdS (2.20) Again, spherical coordinates have been used for dΩand the surface element dS is given by dS= ds1ds2, with thegeneralized coordinates
s=[s1, s2]T. The flux emitted into a detector occupying only a fraction of the surrounding hemisphere can be derived from Eq. (2.20) by inte- grating over the solid angleΩDsubtended by the detector area instead of the whole hemispheric enclosureH.
Inverse square law. A common rule of thumb for the decrease of ir-
radiance of a surface with distance of the emitting source is theinverse square law. Solving Eq. (2.11) for dΦ and dividing both sides by the area dS of the receiving surface, the irradiance of the surface is given by
E= dΦ
dS =I dΩ
2.3 Radiometric quantities 25
θ
Io
I coso θ
Figure 2.8:Illustration of angular distribution of radiant intensity emitted from a Lambertian surface.
For small surface elementsdS perpendicular to the line between the point source and the surface at a distancer from the point source, the subtended solid angle dΩcan be written as dΩ= dS/r2. This yields the expression
E= IdS
dSr2 = I
r2 (2.22)
for the irradianceE at a distancer from a point source with radiant intensityI. This relation is an accurate and simple means of verifying the linearity of a detector. It is, however, only true for point sources. For extended sources the irradiance on the detector depends on the geometry of the emitting surface (Section2.5).
Lambert’s cosine law. Radiant intensity emitted from extended sur-
faces is usually not evenly distributed in angle. A very important rela- tion for perfect emitters, or perfect receivers, isLambert’s cosine law. A surface is calledLambertian if its radiance is independent of view angle, that is, L(x,θ,φ) =L(x). The angular distribution of radiant intensity can be computed directly from Eq. (2.18):
I(θ)=cosθ S
L(x)dS=I0cosθ (2.23)
It is independent of angleφ and shows a cosine dependence on the angle of incidenceθas illustrated in Fig.2.8. The exitance of a planar Lambertian surface is derived from Eq. (2.14), pullingLoutside of the angular integrals M(x)=L(x) 2π 0 π/2 0 cosθsinθdθdφ=πL(x) (2.24)
The proportionality factor ofπ shows that the effect of Lambert’s law is to yield only one-half the exitance, which might be expected for a sur- face radiating into 2π steradians. For point sources, radiating evenly into all directions with an intensityI, the proportionality factor would be 2π. Non-Lambertian surfaces would have proportionality constants smaller thanπ.
Another important consequence of Lambert’s cosine law is the fact that Lambertian surfaces appear to have the same brightness under all view angles. This seems to be inconsistent with the cosine dependence of emitted intensity. To resolve this apparent contradiction, radiant power transfer from an extended source to a detector element with an area of finite size has to be investigated. This is the basic topic of radiometry and will be presented in detail in Chapter4.
It is important to note that Lambert’s cosine law only describes per- fect radiators or perfect diffusers. It is not valid for real radiators in general. For small angles of incidence, however, Lambert’s law holds for most surfaces. With increasing angles of incidence, deviations from the cosine relationship increase (Section2.5.2).
2.3.5 Spectral distribution of radiation
So farspectral distributionof radiation has been neglected. Radiative flux is made up of radiation at a certain wavelengthλ or mixtures of wavelengths, covering fractions of the electromagnetic spectrum with a certain wavelength distribution. Correspondingly, all derived radio- metric quantities have certain spectral distributions. A prominent ex- ample for a spectral distribution is the spectral exitance of a blackbody given by Planck’s distribution [CVA1, Chapter 2].
LetQbe any radiometric quantity. The subscriptλdenotes the cor- respondingspectralquantityQλconcentrated at a specific wavelength within an infinitesimal wavelength intervaldλ. Mathematically,Qλ is defined as the derivative ofQwith respect to wavelengthλ:
Qλ= ddQλ = lim
∆λ→0
∆Q
∆λ (2.25)
The unit ofQλis given by[·/m]with[·]denoting the unit of the quan- tityQ. Depending on the spectral range of radiation it sometimes is more convenient to express the wavelength dependence in units of [·/µm](1µm = 10−6m) or [·/nm](1 nm = 10−9m). Integrated quan- tities over a specific wavelength range [λ1, λ2] can be derived from spectral distributions by Qλ2 λ1= λ2 λ1 Qλdλ (2.26)