Photometry is the science of light measurement in a manner proportional
to human visual response according to an accepted average human visual
response function. The internationally accepted standard in this regard is
the CIE spectral luminous efficiency function for photopic vision, usually
designated as V(λ) [2,
34]. This function is shown as the
curve labeled "green" in Fig. 7. The other curves plotted in Fig. 7
are the CIE color matching functions "red" and "blue" used to define points in
the CIE color coordinate system [2,
35]. The definition of the candela and
the V(λ) function provide a means of coupling the photometric
and radiometric units shown in Table 1 in the manner described below.
Figure 7. CIE color matching functions. The curve labeled "green" is the spectral luminous efficiency function for photopic vision, V(λ).
Let Qλ represent the general radiometric quantity on on the left side of Table 1 and Qv represent the general photometric equivalent function, then,
Km is the luminous efficacy defined by the CIPM to be 683 lm/W for photopic vision . The range of the integral has practical limits restricted to the region of non-zero values for V(λ). Figure 8 illustrates a typical measurement configuration for a light source and filter detector system. A source is placed a distance r from the detector system whose input is determined by the amount of optical power entering the precision aperture of area A. In most configurations in photometry and radiometry one would choose the distance r large compared to the dimensions of the aperture and the source size. For utmost precision one can consider the appropriate configuration factors for the total system of source and receiver. For our purposes we shall assume the limit where the source size is approximated by a point source, the aperture is sufficiently small so that there are no variations in the light flux distribution over the aperture, and the responsivity is appropriately area averaged in order to avoid integration over the aperture.
Figure 8. Schematic of a setup for using an absolutely calibrated photometer to perform luminous intensity and illuminance measurements. The distance r, is large compared to the size of the lamp and the dimensions of the photometer's aperture. The distance r, is measured from the lamp filament to the aperture of the photometer.
The photometer is calibrated on the DSC and has an absolute spectral responsivity given by s(λ) [A/W]. The uniform irradiance of the source Eλ(λ) on the aperture Aproduces a current i0 from the detector systems,
The filter is designed such that its transmittance combined with the responsivity of a photodetector produces a response of the FR nearly proportional to the V(λ) function. An example of such a detector's response is indicated as FR#2 in Fig. 6.
The responsivity of the photometer to the light in terms of luminous flux can be defined as shown in the equation,
where Rv,f is the photometric response of the system to luminous flux measured in terms of A/lm. Eliminating i0 between eqs. (2) and (3) results in
To the extent that s(λ) and V(λ) have the same functional form, eq. (4) reduces to having the lumen response determined in terms of the ratio of the absolute response at 555 nm and the luminous efficacy. This is evident if one writes,
where sn(λ) is the spectral responsivity normalized to the value at 555 nm.
The luminous flux responsivity then can be written,
If the functional forms of sn(λ) and V(λ) are sufficiently close one can write,
Knowing Eλ(λ), the correction terms can be calculated for the spectral distribution of various sources. These matters are discussed extensively in the technical literature and will not be reproduced here [2, 36].
Assuming the luminous flux is Φ uniform over the aperture A, we can write,
|Φv = Ev · A,||(8)|
where Ev is the illuminance and conclude that the illuminance responsivity Rv,i is,
In the approximation that the source is a point source the illuminance is related to the luminous intensity by well known inverse square law,
We can write the luminous intensity Iv expressed in candela as,
Keeping in mind that Rv,f was determined by a direct measurement on the DSC utilizing eqs. (6) and (7), it is noted that the SI unit of the candela is derived directly from the primary detector standard and that the auxiliary unit for illuminance in lux or lm/m2 is similarly derived. Employing the wide dynamic range of photopically corrected detector systems, a single calibrated instrument can serve to establish luminous intensity and illuminance units over a broad range and removes the necessity of every calibration laboratory maintaining a large inventory of lamps to provide complete calibration coverage .
Light sources used for illumination are rated in terms of the total luminous flux output, often abbreviated to luminous flux, measured in lumens. The measurement of the luminous flux involves the integration of the luminous intensity of the source over the total solid angle of illumination, ordinarily 4π sr. This is usually accomplished with a goniophotometer which uses a calibrated photometer in a mechanical arrangement to move the photometer over the solid angle of interest . This method lends itself to immediate application of standard detectors because a photometer can be directly calibrated as an illuminance meter and upon geometric considerations and suitable summation of signals, the luminous flux is directly deduced.
A second frequently used integration method is the use of an integrating sphere for averaging the output of a particular source and deriving its luminous flux by comparison to a known source in the same sphere. This method relies upon the availability of a luminous flux standard source and the ability to make appropriate corrections for source configuration issues. NIST has recently developed a new method utilizing an integrating sphere but relying upon the comparison to a known externally provided luminous flux introduced into the sphere by a measured illuminance within a defining aperture [38, 39].
Figure 9 illustrates the main features of the NIST integrating sphere approach to a direct detector measurement of the luminous flux. A critical aspect of this approach has been the development of techniques to characterize the integrating sphere for effects from sources that supply luminous flux in different manners. An illuminance or irradiance standard lamp is used as a source of external radiation with the only requirements on its performance being stability of output and operation at an identified color temperature. A precision aperture of area A defines the amount of luminous flux that will enter the sphere through an opening.
Figure 9. Schematic diagram of the integrating sphere method NIST uses to determine luminous flux based upon the detector candela.
The standard illuminance meter measures the illuminance Ev behind the aperture, and assuming the flux is homogeneous in space, the total flux Φvin entering the sphere is,
|Φvin = Ev · A [lm] .||(12)|
The monitor photometer or spectral radiometer records the signal as a result of supplying the known flux. The test lamp is then placed into operation and the external source eclipsed with a shutter. A new reading on the monitor allows for determination of the luminous flux of the test lamp by the comparative equation,
|Φvtest = Φvin · (monitor signal ratio) · (corrections) .||(13)|
The corrections depend upon angular correction factors for light impinging upon different regions of the sphere and for the fact that the external light impinges upon the sphere at an oblique angle whereas the test lamp provides illumination at nearly normal incidence. Other corrections can occur for color temperature variations between the external source and test lamp. All these corrections are small, with the largest on the order of 1% and smallest on the order of a few tenths of a percent. Ohno has written a detailed analysis of the correction strategy and the reader is referred to the original literature for the details [13, 40]. NIST believes that the luminous flux unit can be maintained in this manner to a relative combined standard uncertainty of 0.3%. The wide dynamic range of the photometer allows for a characterization of the sphere and the calibration of a lamp at a given lumen level to be transferred to other lamps over a large range of lumen values.
As a practical matter for calibration purposes it is convenient to calibrate a selection of standard lamps using the absolute method described above. These standard lamps can be used to calibrate other lamps by ordinary substitution methods employing the sphere. Periodically the sphere must be checked and the standard lamp calibrations verified. The characterization of the sphere described in the references is a straightforward procedure and can be duplicated in most well equipped photometry laboratories and provides a mechanism for other laboratories to establish a luminous flux unit based upon calibrated photometers. A single calibrated photometer could then be used to assure the maintenance of a number of photometric quantities with good stability and accuracy. It might be necessary to have periodic measurement verifications provided by NIST to ensure that the procedures developed in an individual laboratory are correct and maintained over time. Implementing these procedures could result in considerable savings to calibration laboratories by not maintaining large selections of calibration artifacts. NIST encourages its major customers to consider this strategy for incorporation into their long term planning. An update to the NIST SP 250 document that describes the photometric calibration program at NIST and which contains many details of how to establish the detector based photometric program is in preparation .
An extensive discussion of colorimetry is beyond the scope of this Technical Note. Hence the discussion will be limited to the strategy for measurements based upon detectors. Colorimetry consists of measuring either direct, reflected, or transmitted radiation with sensor systems whose responses are weighted as shown in Fig. 7. The result is a set of numbers which define a point in a suitably chosen color coordinate space. Detector filter systems can be constructed that give relative responses that are proportional to the curves in Fig. 7 and can be calibrated on the DSC in the same manner as the photometer. In fact the "green" curve in Fig. 7 is the V(λ) curve. Appropriate corrections for errors introduced by differing source spectral distributions and other effects can be determined to provide a strategy to utilize the DSC calibrated detectors to maintain colorimetric measurements. This project is in its early stages at NIST and will be reported on at an appropriate time.