## IV.  Filter Detector Systems and Use

 (14)

where L is the radiance of the wide aperture source, d is the distance between the midpoints of the apertures and the θs are the angles defining the inclination of the apertures with respect to the central ray passing through the midpoints of the area elements [45]. These concepts can be generalized for apertures of an arbitrary size by considering the limit of small areas in eq. (14) and summing contributions from the elements of area using integration techniques. In this connotation the areas A1 and A2 become elements of area of larger apertures at the positions indicated in Fig. 10. Using this idea and letting the elements of area become differentials, the total flux Phi exiting aperture A2 can be determined by integrating the radiance over the two apertures,

 (15)

Figure 10.  Diagram illustrating relationships for the fundamental concepts in the measurement of radiance and optical power. The vertical dotted lines represent two reference planes separated by a distance, d. Two apertures, A1 and A2, determine the geometry of the arrangement and the amount of optical power incident upon the detector from the wide aperture source.

In the general circumstance this is a complicated integration because the radiance may vary over the aperture and the distance d is a function of the position on the apertures. Our application provides some simplifying and useful conditions which make the problem tractable in mathematical form. The sources to be employed for radiance and irradiance measurements are Lambertian sources and have sufficiently large extent to completely overfill the aperture A1. Additionally the detector has a diameter sufficient to intercept all the radiation exiting through aperture A2. This field of view is delineated by the extreme rays shown in Fig. 10. With these assumptions, L is constant and can be removed from the integrals. The result is often conveniently written;

 (16)

where F1→ 2 is the configuration factor defined for the two apertures [45, 46]. For the circumstance we are exploring, F1→ 2 is given by the following:

 (17)

where r1 and r2 are the radii of the apertures A1 and A2 respectively. It is convenient to rewrite eq. (16) in terms of the variables, r1, r2, and d, to obtain insight into its application.

 (18)

The quantity (r12 + r22 + d2) can be factored out and since d is much larger than the radii of the apertures, the expression can be expanded using the binomial expansion with the following result:

 (19)

The major functional relation in eq. (19) can be seen to evolve directly from eq. (14) if the angles are assumed small and the distance between the two apertures is large compared to the dimensions of the apertures. For this discussion it is assumed that the apertures are large compared to the wavelength of light being used and hence diffraction and interference effects can be ignored. In some cases in radiometry and photometry this approximation may not be valid.

In the NIST implementation of the direct radiance determination, the radii r1 and r2 are 3 mm and 2 mm respectively and d is 500 mm: hence the first correction in the brackets, in eq. (19), is on the order of 1.4 × 10-4. For most applications with these types of dimensions the radii squared terms can be neglected with respect to d2 in the denominator terms. The variable temperature blackbody source is designed to provide a spatially uniform beam and hence we can write the irradiance in the plane of aperture 2 as,

 (20)

The same equations govern the respective spectral quantities where the radiance L is replaced by the spectral radiance Lλ. Detector systems shown in Figs. 5 and 6 can be used to deduce the temperature of the wide aperture variable temperature blackbody source and hence establish units of spectral radiance and irradiance and radiation temperature. This technique relies upon knowing the aperture areas to at least the intended accuracy of the measurement or as a practical matter, somewhat better than the desired accuracy. To assist in the achievement of the highest accuracy in these measurements, NIST has developed a new facility to characterize apertures employed for radiometric and photometric purposes. The facility features the capability of aperture area measurement with a relative combined standard uncertainty of 0.04% [47].

Figure 11.  Schematic diagram of NIST apparatus to determine scales of spectral radiance and irradiance. The linear dimensions are not to scale in order to better show the details of the measurement. The wide aperture blackbody source has an opening of 17 mm and is placed approximately 60 mm behind the first aperture. The 10 mm square detector is approximately 43 mm behind aperture A2.

The NIST system designed to determine the spectral radiance and irradiance units is shown schematically (distances not to scale) in Fig. 11. The configuration meets the criteria for overfilling aperture A1 and having sufficient aperture to collect all the light exiting aperture A2. The spectral responsivity of the detector, s(λ), is determined and the responsivity spatial uniformity characterized in the DSC. It is important to understand the spatial uniformity since the detector is underfilled and in some cases detectors have shown position sensitive responsivity. In cases where the response variations are large enough to affect the measurement accuracy, the spatial response should be appropriately averaged over the area of the detector to be utilized in the experiment. With these assumptions the output current of the detector system is,

 (21)

The spectral radiance Lλ(λ) and spectral radiant flux Φλ(λ) are known functions of wavelength λ and absolute temperature T and are given by the well discussed Planck radiation law [45,48]. Equation (21) can be numerically solved to find a value of temperature that satisfies the conditions of the equation. The accuracy of the temperature determination is directly related to the accuracy of the determination of geometric quantities, the current, and the FR spectral responsivity, s(λ). It is important to have the FR characterized over the entire wavelength range of sensitivity of the photodiode or other photo-conversion device to account for any out-of-band problems in the filter used. For example, if the filter has significant infrared leakage and the detector is a silicon photodiode, significant errors can result due to the increasing output of thermal sources in the infrared. These issues have been discussed in the literature [42-44]. NIST expects temperature to be determined to within 0.10 K [49]. As a check on stability and to provide redundancy of measurement, several FRs will be used to determine the temperature and to monitor the calibration procedures.

The spectral irradiance Eλ(λ) can be obtained by examination of eq. (20) when it is rewritten in the spectral form,

 (22)

When the spectral irradiance is measured at NIST, an integrating sphere is used in conjunction with a known aperture and a monochromator. Hence the aperture A2 designated in eq. (22) is not necessarily the same one used in the temperature determination but it must meet the same dimensional requirements discussed above in terms of geometrical constraints. In practice the known spectral irradiance of the blackbody is used to calibrate the combined integrating sphere and monochromator system and determine the spectral irradiance of other sources by relative measurement. The spectral radiance measurements will be made utilizing an imaging system in conjunction with the monochromator system employed for the irradiance work. In normal operation a working standard lamp may be used to transfer from the variable temperature blackbody to calibration test lamps. In this arrangement the FRs can be used to check the stability of the measurement technique with time.

The procedures outlined here provide a mechanism to establish source spectral units in terms of a FRs absolute spectral responsivity. This responsivity is directly traceable to the HACR and geometric configuration factors which can accurately be determined. The same procedures can be carried out by other laboratories to establish and maintain their source spectral units and could result in better stability of the calibration endeavor and perhaps a savings in the purchase and maintenance of source standards.