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FIGURE 3 Illustration of radiometric terms and their defining variables. dΩ, differential solid angle at source; dA, differential area cut off by dΩ on surface of sphere of radius r; dΦ, differential radiant flux; dA_s, differential area of source; θ, angle relative to source normal.

The radiometric and photometric terms relating to radiation received on a surface are summarized in Table 1 (part (b)).

      The irradiance at a distance r from a point source that is emitting a total radiant flux Φ isotropically can be calculated as

(6)

      This relationship is known as the inverse square law. The inverse square law is not valid within close distances of an area source. However, when the distance from an area source is more than five times greater than the longest dimension of the source, the irradiance does become proportional to the inverse square of the distance from the source (10). Under these circumstances, given a Lambertian source of area A_s, the radiant flux dΦ received at a differential receiving area dA_r is

      (7)

      where θ is the angle between the line of sight and the perpendicular to the radiating surface, and θ_r is the angle between the line of sight and the perpendicular to the receiving surface. The solid angle subtended by the differential receiving area is , where the factor cos θ_r takes account of the fact that the receiving area will subtend a smaller solid angle, and so capture less radiant flux, if it is not oriented perpendicular to the line of sight. The irradiance is then

(8)

      which is Lambert's cosine law combined with the cosine response of the receiving surface.

3 SOURCES OF BROADBAND OPTICAL RADIATION

      The purpose of this section is to provide a general overview of the spectral characteristics of common sources of broadband optical radiation: blackbody radiators, gas discharge and arc lamps, fluorescent lamps, electrical discharges, and LEDs.

      3.1 Blackbody Sources

      An ideal blackbody is an object that absorbs all incident radiation. A blackbody reradiates energy with an emitted spectral distribution that depends only on the temperature of the blackbody in accordance with Planck's equation:

(9)

where M_λ is the spectral radiant exitance, k is Boltzmann's constant, and T is the absolute temperature. The emission spectra for blackbodies at various temperatures are shown in Figure 4. As the temperature increases, the peak of the blackbody emission spectrum shifts to shorter wavelengths, and the power emitted at any wavelength increases. A blackbody at room temperature (300 K) emits nearly all of its radiation in the IR region. A blackbody at 3000 K, the temperature of the filament in a tungsten–halogen lamp, has its emission peak in the near IR but emits radiation throughout the visible region and well into the UV. Though no real object is a perfect blackbody emitter, the theoretical blackbody spectral emission curve is a useful approximation of the emission spectra of many optical radiation sources, including the sun, incandescent lamp filaments, quartz tungsten halogen lamps, electrical heating elements, furnaces, molten metals, and other hot objects.