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2.1Definition and measurement of radiometric quantities

B. Wende, J. Fischer

2.1.1 Introduction

Radiometry is the science and technology of the measurement of electromagnetic energy. Here we confine ourselves on the subfield of optical radiometry which covers the measurement of electromagnetic radiation in the wavelength range from about 0.01 µm to 1000 µm. Radiometric quantities are derived from the quantity energy. The corresponding photometric quantities on the other hand involve the additional evaluation of the radiant energy in terms of a defined weighting function, usually the standard photometric observer. In the following only the definitions of the radiometric quantities are explained in detail. Starting from the radiant energy the other fundamental radiometric quantities radiant power, radiant excitance, irradiance, radiant intensity, and radiance are derived by considering the additional physical quantities time, area, and solid angle.

The radiometric quantities defined in abstract terms are practically embodied by radiometric standards. Radiometry is based on primary detector standards and primary source standards. Primary detector standards are mostly electrical-substitution thermal detectors whereas for primary source standards the emitted radiant power is accurately calculable. For the radiometric measurement of cw laser emission radiation detectors or radiometers calibrated against primary detector standards are the preferred secondary standards. The detection principle of the radiometers could be thermal (thermopiles, bolometers, and pyroelectric detectors) or photoelectric (semiconductors). As secondary standards for pulsed laser radiation mostly thermally absorbing glass-disk calorimeters are used. These standards are derived from the cw standards using accurately measured shuttering of the laser radiation to produce pulses of known radiant energy.

2.1.2 Definition of radiometric quantities

Radiometric and photometric quantities are represented by the same principal symbol and may be distinguished by their subscripts. While radiometric quantities either have the subscript “e” or no subscript (as in the whole Chap. 2.1), photometric quantities have the subscript “v”, where “e” stands for “energetic” and “v” for “visible”. The most frequently used radiometric quantities are listed in Table 2.1.1 together with their symbols, defining equations, and units. The additional physical quantities applied in Table 2.1.1 are the time t, the element of solid angle d ω, and the angle θ between the line of sight and the normal of the radiating or receiving surface with the area element dA, see Fig.2.1.1.

In the case that the quantities are functions of wavelength their designations must be preceded by the adjective “spectral”. For example, the symbol for spectral radiance is L(λ). This has to be well distinguished from the convention for the spectral concentration of a quantity, which is also preceded by the adjective “spectral”. In that case, however, the symbol has the subscript λ, i.e. dL/dλ = Lλ.

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Table 2.1.1. Radiometric quantities, their defining equations and units.

To explain the defining equations given in Table 2.1.1 a radiation source of finite extent is considered. If we surround the radiation source with a closed surface and calculate the radiant energy Q penetrating the surface per unit time we get the total radiant power Φ emitted by the source. For clarity, the above mentioned symbols for the spectral properties of the radiation are omitted in this chapter. The radiant power per unit area of the radiation source associated with the emission into the hemispheric space above dA is defined as the radiant excitance M . At this point it is appropriate to introduce the radiation incident from all directions in the hemispheric space above the surface of a detector. The irradiance E is defined as the radiant power incident on a surface per unit area of the surface. The irradiance represents also the energy which propagates per unit time through the unit area perpendicular to the direction of energy transport. This is known as the density of energy flow identical to the magnitude of the Poynting vector averaged over time.

Coming back to the source-based radiometric quantities we consider now the radiant power proceeding from a point source per unit solid angle d ω in a specified direction. The corresponding quantity appropriate especially for nearly point-shaped sources is denoted as radiant intensity I. If we generalize and consider again a source of finite extent the directional nature of radiation has to be taken into account accurately. From Fig. 2.1.1 we formally define as radiance L the radiant power emitted in the (θ, ϕ) direction, per unit area of the surface normal to this direction and per unit solid angle. Note that the area dAn used to define the radiance is the component of dA perpendicular to the direction of the radiation. This projected area is equal to cos θ dA and in e ect, this is how dA would appear to an observer situated on the surface in the (θ, ϕ) direction.

Although the directional distribution of surface emission varies according to the nature of the surface, there is a special case which provides a reasonable approximation for many surfaces. For

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an isotropically di use emitter the radiance is independent of direction:

Such an emitter is denoted as a lambertian radiator which emits in accordance with Lambert’s cosine law:

The radiant intensity of a perfectly di use surface element in any direction varies as the cosine of the angle between that direction and the normal to the surface element. It is noted that this law is consistent with the definitions of radiance and radiant intensity given in Table 2.1.1. It may be helpful to derive the relationship between radiance and radiant excitance for a lambertian radiator. The radiant excitance into the hemispheric space above dA is calculated from the radiance by integration over the solid angle d ω = sin θ dθ dϕ:

By removing L(θ, ϕ) from the integrand according to (2.1.1) and performing the integration we get

Note that the constant appearing in the above expression is π, not 2π, and has the unit steradian (sr).

2.1.3 Radiometric standards

2.1.3.1 Primary standards

Depending on the application primary source and primary detector standards are used to establish radiometric scales. Black-body radiators of known temperature with calculable spectral radiance are operated as primary source standards at temperatures up to about 3200 K [96Sap]. Due to the steep decrease of their Planckian radiation spectrum in the UV spectral range radiometry with black-body radiators is limited to wavelengths above 200 nm. In comparison with a black-body radiator, the maximum of the synchrotron radiation spectrum emitted by an electron storage ring is shifted to shorter wavelengths by several orders of magnitude [96Wen]. In a storage ring electrons move with nearly the velocity of light along a circular trajectory and emit a calculable radiant power through an aperture stop situated near the orbital plane. Radiometry can thus be extended into the X-ray region up to photon energies of 100 k eV.

Electrical-substitution thermal detectors operated at ambient temperature have been the most frequently used primary detector standards. However, their performance is limited by the thermal properties of materials at room temperature resulting in complicated corrections that have to be applied. Hence, their uncertainties remain near 0.1 % to 0.3 % [89Fro, 79Wil]. Cryogenic radiometers have been developed to satisfy the increasing demands for more accurate detector standards from users especially in new and expanding fields of optical fibers, laser technology, and space science. Today, these instruments with absorption cavities at nearly the temperature of liquid helium

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