The spectral radiance and irradiance units have traditionally been based
upon the use of blackbody sources with temperatures defined in terms of
accepted melting or freezing points of materials
[4, 5].
The blackbody sources are characterized to ensure their spectral output can be
correctly described by Planck's radiation law. Recently Mielenz, *et al*.
determined the freezing point of gold using calibrated detectors to determine
the absolute spectral radiance within a selected wavelength band and inferred
the temperature by application of Planck's radiation law
[42]. Using the ideas developed by
Mielenz, *et al.*, NIST has designed a FR based system coupled with a
variable temperature blackbody to maintain the units of spectral radiance and
irradiance [43,
44]. This realization is based upon
concepts associated with the conservation of radiance in a nonabsorbing medium
with a constant index of refraction [14,
45]. Figure 10 illustrates the basic
geometry and ideas associated with the method employed to determine the
irradiance and radiance units. If *A*_{1}and
*A*_{2} are the areas of aperture 1 and aperture 2
respectively, the flux Δ*Φ* emergent through
*A*_{2} can be expressed as

(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 *A*_{1} and
*A*_{2} 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
*A*_{2} 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,A_{1}andA_{2}, 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
*A*_{1}. Additionally the detector has a diameter sufficient to
intercept all the radiation exiting through aperture *A*_{2}. 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 *F*_{1→ 2} is the configuration factor defined
for the two apertures [45,
46]. For the circumstance we are
exploring, *F*_{1→ 2} is given by the following:

(17) |

where *r*_{1} and *r*_{2} are the radii of the
apertures *A*_{1} and *A*_{2} respectively. It is
convenient to rewrite eq. (16) in terms of the variables,
*r*_{1}, *r*_{2}, and *d*, to obtain insight
into its application.

(18) |

The quantity *r*_{1}^{2} +
*r*_{2}^{2} + *d*^{2})*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* r*_{1} and *r*_{2} 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 *d*^{2} 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
*L*_{λ}. 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 A_{2}.

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 *A*_{1}
and having sufficient aperture to collect all the light exiting aperture
*A*_{2}. 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 *L*_{λ}(λ) and spectral
radiant flux *Φ*_{λ}(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 *E*_{λ}(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 *A*_{2} 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.