Radiometry, Photometry, Spectroscopy, and Color

Radiometry Terms

Radiometry is the topic in optics describing and measuring radiation and its interaction with matter. Because of the dual nature of radiation, the radiometric terms refer either to energy or to particles; in case of electromagnetic radiation, the particles are photons (Section 6.2.4). If it is required to distinguish between the two types, the indices e and p are used for radiometric terms.

Radiometry is not a complex subject. It has only become a confusing subject following diff erent, inaccurate, and often even wrong usage of its terms. Moreover, radiometry is taught less frequently and less thor- oughly than other subjects in optics. Thus, knowledge about radiometry is less widespread. However, it is a very important subject for imaging. Geometrical optics only tells us where the image of an object is located, whereas radiometry says how much radiant energy has been collected from an object.

 

Radiant Energy. Since radiation is a form of energy, it can do work. A body absorbing radiation is heated up. Radiation can set free elec- tric charges in a suitable material designed to detect radiation. Radiant energy is denoted by Q and given in units of Ws (joule) or number of particles (photons).

 

Radiant Flux. The power of radiation, i. e., the energy per unit time, is known as radiant fl ux and denoted by Φ:

=

Φ dQ.                                     (6.8)

dt

This term is important to describe the total energy emitted by a light source per unit time. Its unit is joule/s (Js− 1), watt (W), or photons per s (s− 1).

154                                                                             6 Quantitative Visualization

 

a                                                                     b

X                                                  X

 

 

Figure 6.3: a Defi nition of the solid angle. b Defi nition of radiance, the radiant power emitted per unit surface area dA projected in the direction of propagation per unit solid angle Ω.

 

Radiant Flux Density. The radiant fl ux per unit area, the fl ux density, is known by two names:

irradiance E = d Φ  ,       excitance M = d Φ  .                   (6.9)

dA₀                                     dA₀

The irradiance, E, is the radiant fl ux incident upon a surface per unit area, for instance a sensor that converts the radiant energy into an elec- tric signal. The unit of irradiance is W m− 2, or photons per area and time (m− 2s− 1). If the radiation is emitted from a surface, the radiant fl ux density is called excitance or emittance and denoted by M.

=

Solid Angle. The concept of the solid angle is paramount for an under- standing of the angular distribution of radiation. Consider a compact source at the center of a sphere of radius R beaming radiation outwards in a cone of directions (Fig. 6.3a). The boundaries of the cone outline an area A on the sphere. The solid angle (Ω ) measured in steradians (sr) is the area A divided by the square of the radius (Ω A/R2). Although the steradian is a dimensionless quantity, it is advisable to use it explic- itly when a radiometric term referring to a solid angle can be confused with the corresponding non-directional term. The solid angle of a whole sphere and hemisphere are 4π and 2π, respectively.

Radiant Intensity. The (total) radiant fl ux per unit solid angle emitted by a source is called the radiant intensity I:

=

I  d Φ  .                                                (6.10)

dΩ

It is obvious that this term only makes sense for describing compact or point sources, i. e., when the distance from the source is much larger

6.3 Radiometry, Photometry, Spectroscopy, and Color                   155

than its size. This region is also often called the far-fi eld of a radiator. Intensity is also useful for describing light beams.

Radiance. For an extended source, the radiation per unit area in the direction of excitance and per unit solid angle is an important quantity (Fig. 6.3b):

L    d2Φ

d2Φ

 

= dA dΩ = dA₀ cos θ dΩ .                                    (6.11)

=    ·

The radiation can either be emitted from, pass through, or be incident on the surface. The radiance L depends on the angle of incidence to the surface, θ (Fig. 6.3b), and the azimuth angle φ. For a planar surface, θ and φ are contained in the interval [0, π /2] and [0, 2π ], respectively. It is important to realize that the radiance is related to a unit area in the direction of excitance, dA dA₀ cos θ. Thus, the eff ective area from which the radiation is emitted increases with the angle of incidence. The unit for energy-based and photon-based radiance are W m− 2sr− 1 and s− 1m− 2sr− 1, respectively.

Often, radiance — especially incident radiance — is called brightness. It is better not to use this term at all as it has contributed much to the confusion between radiance and irradiance. Although both quantities have the same dimension, they are quite diff erent. Radiance L describes the angular distribution of radiation, while irradiance E integrates the radiance incident to a surface element over a solid angle range corre- sponding to all directions under which it can receive radiation:

 

∫

∫

∫

π /2 2π

E = L(θ, φ ) cos θ dΩ =

Ω                                     0

L(θ, φ ) cos θ sin θ dθ dφ.                  (6.12)

0

The factor cos θ arises from the fact that the unit area for radiance is related to the direction of excitance (Fig. 6.3b), while the irradiance is related to a unit area parallel to the surface.

 

Spectroradiometry

Because any interaction between matter and radiation depends on the wavelength or frequency of the radiation, it is necessary to treat all ra- diometric quantities as a function of the wavelength. Therefore, we de- fi ne all these quantities per unit interval of wavelength. Alternatively, it is also possible to use unit intervals of frequencies or wave numbers. The wave number denotes the number of wavelengths per unit length interval (see Eq. (2.14) and Section 2.3.4). To keep the various spectral quantities distinct, we specify the dependency explicitly, e. g., L(λ ), L(ν ), and L(k).

156                                                                             6 Quantitative Visualization

 

The radiometric terms discussed in the previous section measure the properties of radiation in terms of energy or number of photons. Pho- tometry relates the same quantities to the human eyes’ response to them. Photometry is of importance to scientifi c imaging in two respects: First, photometry gives a quantitative approach to radiometric terms as they are sensed by the human eye. Second, photometry serves as a model for how to describe the response of any type of radiation sensor used to convert irradiance into an electric signal. The key in understanding pho- tometry is to look at the spectral response of the human eye. Otherwise, there is nothing new to photometry.

 

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