The radiant flux from a light source varies with the source and the wavelength. Understanding the variations is key to lighting design.
A physicist studying a system radiating electromagnetic energy would use instruments that measure that energy in the usual MKS units. The energy emitted by the system, U, would be given in joules. The time rate of production of radiant energy, radiant flux P, would be defined as:
where t is time. The MKS unit of radiant flux would be, therefore, joules per second or watts.
Generally, the radiant flux from a source differs for different wavelengths. The flux at a given wavelength, λ, and in a spectral interval dλ is given by Pλdλ. A plot of Pλ versus λ gives the power spectrum of the radiator. Power spectra may be either continuous or discrete. The usual tungsten filament lamp has a continuous power spectrum like that of Figure 1. The mercury lamp spectrum shown in Figure 2 is a typical line spectrum with the spectral lines corresponding to energy level transitions. Fluorescent lamps have discrete spectra superimposed on a continuous spectra. The total radiant flux P emitted by a light source is the integral of Pλdλ over all wavelengths,
This is the area under the Pλ curve.
Figure 1: A typical continuous power spectrum.
Figure 2: A typical line spectrum, of a mercury vapor lamp.
Because visible light occupies only a small part of the electromagnetic spectrum and because the sensitivity of the eye varies considerably with the actual wavelength of the visible light, the units quantifying light energy in vision science are different from those used in physics. These units are based on the experimentally measured spectral sensitivity of the eye quantified by the spectral sensitivity function Vλ, or the relative spectral sensitivity function vλ. One way to determine vλ, in principle at least, would be to have a typical observer compare the amount of energy required to make the light sensation or “brightness” of light at one wavelength match that of light at another wavelength. If, for example, three times as much radiant energy were required at 500 nm as at 540 nm to produce the same magnitude of visual sensation, then:
(In practice, the color differences in the two stimuli confound this type of experiment, and more elaborate techniques are necessary. The exact results vary from observer to observer and with the state of light adaptation of the observer, but we’ll ignore those effects here. The vλ curve accepted as standard is actually an average over a great many normal observers.)
The relative spectral sensitivity function is given by the graph of Figure 3 or the numerical tabulation of Table 1. The spectral sensitivity function vλ is proportional to itself according to the equation:
The value of the constant has been re-established from time to time so some texts may use 680 while others use 685. The value 683 is the currently accepted value.
Figure 3: The relative spectral luminosity curve, vλ.
λ(nm)
vλ
λ(nm)
vλ
λ(nm)
vλ
λ(nm)
vλ
400
0.000
500
0.323
600
0.631
700
0.004
410
0.001
510
0.503
610
0.503
710
0.002
420
0.004
520
0.710
620
0.381
720
0.001
430
0.012
530
0.862
630
0.265
730
0.000
440
0.023
540
0.954
640
0.175
—
—
450
0.038
550
0.995
650
0.107
—
—
460
0.060
560
0.995
660
0.061
—
—
470
0.091
570
0.952
670
0.032
—
—
480
0.139
580
0.870
680
0.017
—
—
490
0.208
590
0.757
690
0.008
—
—
Table 1: A tabulation of the values of vλ.
The lumen (abbreviated lm) in the previous equation is a unit defined to characterize luminous flux in the same way the watt characterizes radiant energy. The luminous flux Fλ at wavelength λ in a range dλ is related to the radiant flux in that interval by:
The total luminous flux F is obtained by integrating the above equation to obtain:
The integral is carried out in the range from 410 nm to 720 nm since that is the non-vanishing range of vλ .
In practice the integral in equation (1) is always performed numerically using Simpson’s rule. This means that (1) is approximated by the sum:
Usually λ is taken to be 10 nm.
Values of Pλ and Vλ can be taken from a table like table 1 or interpolated from graphs like those in Figures 1-3.
