t, time; Ω, solid angle at source; As, area of source; θ, angle relative to source normal; Ar, area of receiving surface; J, joule; lm, lumen; s, second; W, watt; sr, steradian; cd, candela; m, meter; lx, lux.
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; dAs, 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
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 As, the radiant flux dΦ received at a differential receiving area dAr is
(7)
where θ is the angle between the line of sight and the perpendicular to the radiating surface, and θr 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
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:
where Mλ 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.
FIGURE 4 Spectral radiant exitance (W m−2 μm−1) in the optical radiation region (100 nm–1000 μm) calculated using Planck's formula for blackbody radiance at different absolute temperatures: 300 K ∼room temperature, 1000 K ∼red‐hot object, 3000 K ∼incandescent lamp filament, 5780 K ∼effective temperature of the sun. The visible radiation band (0.4–0.78 μm) lies between the vertical dotted lines.
3.2 Solar Radiation
The solar radiation spectrum incident on the outermost part of the earth's atmosphere is fairly close to the theoretical blackbody radiation curve for an object at 5780 K, with peak emission around 500 nm in the visible region and significant emission throughout the UV region. Oxygen (O2) strongly absorbs UV at wavelengths shorter than 200 nm, with weaker absorption out to 245 nm, and ozone (O3) in the stratosphere absorbs UV over the range 230–300 nm. Due to atmospheric absorption by O2 and O3, practically no solar UV of wavelength less than 290 nm reaches the earth at sea level. Selected wavelength bands in the visible and IR regions are attenuated by O2, O3, water vapor, and carbon dioxide (11). The spectral distribution of direct sunlight reaching the earth's surface is further altered by Rayleigh scattering, which takes some light out of the direct path from the sun and disperses it around the upper atmosphere. Short wavelengths, including UV and blue light, are more strongly scattered than longer wavelengths. The blue color of the sky comes from this selective scattering of short‐wavelength sunlight. Change in the path length of the sun's rays through the atmosphere throughout the day, as well as variation in path length