**4.2.5 Spectral mismatch correction factor (** \* *F* **)**

The definitions of photometric and colorimetric quantities given above require knowledge of the spectral distribution of the radiation that we are measuring. Most of the instruments in daily use for photometric and colorimetric measurements are simple devices that do not contain the spectroradiometers necessary for the measurement of spectral distributions. The simplest photometers and colorimeters are constructed from detectors, usually of silicon, and transmitting color glass filters. The types and thicknesses of the 3-to-4 types of glasses are chosen such that the relative spectral responsivity of the combination of the detector plus filter is as close to the desired photometric or colorimetric functions (e.g. V(), V'(), *x*( ) , *y*( ) or *z*( ) ) as possible for the construction cost involved. These devices produce a single-number result for the measurement. Since the output of a detector is a voltage or current, this value must be converted to the corresponding photometric or colorimetric quantity by a calibration of the device, which is usually performed by a measurement of a known amount of the desired quantity. This produces a calibration factor that is often built into the electronics of the device.

These CIE spectral functions are not easy to reproduce in this physical form and there will always be errors or uncertainties in measurement using these devices. An estimate of the error involved can be determined by calculating what is called a Spectral Mismatch Correction Factor ( \* *F* ). To derive this factor we compare the quantity that we wish to measure with the quantity that we are actually measuring, using illuminance as an example as shown in Figure 12. The photometer in this case is called an illuminance meter or luxmeter.

Fig. 12. Basic measurement and calibration configuration

The variables that describe the measurement shown in Figure 12 are:

Required value: <sup>T</sup> *<sup>E</sup>*<sup>v</sup> , the illuminance produced by the test source. Known value: <sup>S</sup> *<sup>E</sup>*<sup>v</sup> , the illuminance produced by the standard source. Measured values: <sup>S</sup> *i* , the photometer output for the standard source. <sup>T</sup> *i* , the photometer output for the test source.

Note that the output of the photometer is a single number.

The definitions of photometric and colorimetric quantities given above require knowledge of the spectral distribution of the radiation that we are measuring. Most of the instruments in daily use for photometric and colorimetric measurements are simple devices that do not contain the spectroradiometers necessary for the measurement of spectral distributions. The simplest photometers and colorimeters are constructed from detectors, usually of silicon, and transmitting color glass filters. The types and thicknesses of the 3-to-4 types of glasses are chosen such that the relative spectral responsivity of the combination of the detector plus filter is as close to the desired photometric or colorimetric functions (e.g. V(), V'(),

single-number result for the measurement. Since the output of a detector is a voltage or current, this value must be converted to the corresponding photometric or colorimetric quantity by a calibration of the device, which is usually performed by a measurement of a known amount of the desired quantity. This produces a calibration factor that is often built

These CIE spectral functions are not easy to reproduce in this physical form and there will always be errors or uncertainties in measurement using these devices. An estimate of the error involved can be determined by calculating what is called a Spectral Mismatch Correction Factor ( \* *F* ). To derive this factor we compare the quantity that we wish to measure with the quantity that we are actually measuring, using illuminance as an example as shown in Figure 12. The photometer in this case is called an illuminance meter or

*d*

) as possible for the construction cost involved. These devices produce a

Photometer (detector)

Input Aperture of Area A

*i* , the photometer output for the standard source.

*i* , the photometer output for the test source.

**4.2.5 Spectral mismatch correction factor (** \* *F* **)** 

*x*( ) , *y*( ) 

luxmeter.

Source Standard (S) or Test (T)

Measured values: <sup>S</sup>

<sup>T</sup>

Fig. 12. Basic measurement and calibration configuration

Note that the output of the photometer is a single number.

The variables that describe the measurement shown in Figure 12 are:

Required value: <sup>T</sup> *<sup>E</sup>*<sup>v</sup> , the illuminance produced by the test source. Known value: <sup>S</sup> *<sup>E</sup>*<sup>v</sup> , the illuminance produced by the standard source.

 or *z*( ) 

into the electronics of the device.

We use the material properties of the source output and the photometer components to determine the origin of our measured photometer outputs. The variables are:


*R*( ) , the relative spectral responsivity of the photometer.

The geometrical factors that convert these relative values to absolute values may be written as *a*, *b* and *r* respectively. The photometer outputs for the two sources can then be written:

$$\begin{aligned} \mathbf{h}^{\mathrm{T}} &= \boldsymbol{a} \cdot \boldsymbol{r} \cdot \int\_{\text{all wavelengths}} \boldsymbol{P}\_{\text{e}}^{\mathrm{T}}(\mathcal{L}) \cdot \boldsymbol{R}(\mathcal{L}) \cdot \mathbf{d}\mathcal{L} \\ \mathbf{h}^{\mathrm{S}} &= \boldsymbol{b} \cdot \boldsymbol{r} \cdot \int\_{\text{all wavelengths}} \boldsymbol{P}\_{\text{e}}^{\mathrm{S}}(\mathcal{L}) \cdot \boldsymbol{R}(\mathcal{L}) \cdot \mathbf{d}\mathcal{L} \end{aligned} \tag{31}$$

The measurements we require are the two illuminances:

$$\begin{aligned} E\_{\rm v}^{\rm T} &= a \cdot \upsilon \cdot \int\_{360 \text{ nm}}^{830 \text{ nm}} P\_{\rm e}^{\rm T}(\lambda) \cdot V(\lambda) \cdot \text{d}\lambda\\ E\_{\rm v}^{\rm S} &= b \cdot \upsilon \cdot \int\_{360 \text{ nm}}^{830 \text{ nm}} P\_{\rm e}^{\rm S}(\lambda) \cdot V(\lambda) \cdot \text{d}\lambda \end{aligned} \tag{32}$$

Note that V() is exactly zero outside the 360-830 nm range, whereas the sources and the photometer might very well be contributing a signal outside this range. By taking ratios of the equations we may eliminate the geometrical constants and scaling factors *a*, *b*, r and *v* to obtain the desired illuminance of the test source:

$$\begin{split} \mathbf{E}\_{\mathbf{v}}^{\mathrm{T}} &= \frac{\mathbf{E}\_{\mathbf{v}}^{\mathrm{S}}}{\mathbf{i}^{\mathrm{S}}} \cdot \mathbf{i}^{\mathrm{T}} \cdot \frac{\int\_{360\,\mathrm{nm}}^{830\,\mathrm{nm}} P\_{\mathbf{e}}^{\mathrm{T}}(\boldsymbol{\lambda}) \cdot \mathbf{V}(\boldsymbol{\lambda}) \cdot \mathbf{d}\boldsymbol{\lambda}}{\int\_{\mathrm{all\,\mathrm{wavelength}}}^{\mathrm{T}} P\_{\mathbf{e}}^{\mathrm{T}}(\boldsymbol{\lambda}) \cdot \mathbf{R}(\boldsymbol{\lambda}) \cdot \mathbf{d}\boldsymbol{\lambda}} \cdot \frac{\int\_{\mathrm{all\,\mathrm{wavelength}}}^{\mathrm{S}} P\_{\mathbf{e}}^{\mathrm{S}}(\boldsymbol{\lambda}) \cdot \mathbf{R}(\boldsymbol{\lambda}) \cdot \mathbf{d}\boldsymbol{\lambda}}{\int\_{360\,\mathrm{nm}}^{830\,\mathrm{nm}} P\_{\mathbf{e}}^{\mathrm{S}}(\boldsymbol{\lambda}) \cdot \mathbf{V}(\boldsymbol{\lambda}) \cdot \mathbf{d}\boldsymbol{\lambda}} \\ &= \mathbf{C} \mathbf{\bar{r}} \cdot \mathbf{i}^{\mathrm{T}} \cdot \mathbf{F}^{\mathrm{}} \end{split} \tag{33}$$

where CF is a photometer Calibration Factor that is stored in the electronics of the photometer when the photometer is calibrated with the standard source. This CF converts the measured T *<sup>i</sup>* into <sup>T</sup> *<sup>E</sup>*v when the test source is measured. However, the \* *<sup>F</sup>* term is not included in the photometer measurement or calibration. It can only be determined at the point of measuring the test source, since <sup>T</sup> <sup>e</sup>*P* ( ) is required. In general, <sup>T</sup> <sup>e</sup>*P* ( ) , <sup>S</sup> <sup>e</sup>*P* ( ) and *R*( ) are not known for most photometers and measurements. A standard source that approximates CIE Source A is usually used for the calibration of photometers. The CIE functions are defined, and generic values for <sup>T</sup> <sup>e</sup>*P* ( ) , <sup>S</sup> <sup>e</sup>*P* ( ) and *R*( ) may be used to obtain an estimate of the \* *F* for the photometer or colorimeter and radiation test source in use. The \* *F* can approach 1.0 under one or both of two conditions: 1) if the relative spectral distributions of the standard source and the test source are the same or proportional, i.e. T S e e *P P* () () . This equivalence is one of the bases for the incentive in measurement or calibration to compare 'like-with-like'. 2) if the relative spectral responsivity of the photometer is proportional to the CIE function (V() in our example above). This requires some effort on the part of manufacturers and photometers/colorimeters of many different qualities and cost are available. The evaluation of the quality of photometers and colorimeters has been documented by the CIE in various publications (CIE 179:2007, CIE 069:1987).

#### **5. Measurements and calibrations**

In Section 2 we noted that one of the key parts of a measurement process was to compare two quantities, the quantity we wish to measure and the quantity unit. For most measurements the quantity unit is embodied in a calibrated measurement standard whose quantity value is a multiple or submultiple of the base unit. For example, a standard luminous intensity lamp may produce a luminous intensity of 219.2 cd. To measure the luminous intensity output of another (test) light source, we will need some means of comparing the output luminous intensity of the test source with that of the standard source. This involves the introduction of a third device, that of a transfer device. In our example, this device must be able to compare the luminous intensity of both sources and return values for each lamp that are proportional to the luminous intensity of the lamp, or a value that is the ratio of the luminous intensities of the two sources.

For accurate measurements, it is imperative that the transfer device be capable of comparing the exact quantity that we wish to measure. This includes both the geometrical aspects and the spectral aspects that we discussed in Section 4. The general schematic for this process is shown in Figure 13.

Fig. 13. Measurement Transfer Device

The spectral resolution component indicated for the transfer device is not necessarily the only spectrally important component. In the case of photometric and colorimetric measurements the spectral responsivity of the complete transfer device must be equal to the desired CIE function. If this is not the case, an \* *F* will need to be determined as indicated in Section 4.2.5 above.

If the transfer device is only used to measure the output of the test source by comparison with the standard source, the measured value of the test source is simply:

$$\mathbf{S}^{\rm T} = \frac{\dot{\mathbf{i}}^{\rm T}}{\dot{\mathbf{i}}^{\rm S}} \cdot \mathbf{S}^{\rm S} = \frac{\mathbf{S}^{\rm S}}{\dot{\mathbf{i}}^{\rm S}} \cdot \dot{\mathbf{i}}^{\rm T} \tag{34}$$

where <sup>T</sup> *S* and <sup>S</sup> *S* are the desired quantities for the test source and the standard source, and T *i* and <sup>S</sup> *i* are the signals from the transfer device when measuring these sources. An \* *F* is to be applied to Equation (34) as necessary. If the standard source is a measurement

In Section 2 we noted that one of the key parts of a measurement process was to compare two quantities, the quantity we wish to measure and the quantity unit. For most measurements the quantity unit is embodied in a calibrated measurement standard whose quantity value is a multiple or submultiple of the base unit. For example, a standard luminous intensity lamp may produce a luminous intensity of 219.2 cd. To measure the luminous intensity output of another (test) light source, we will need some means of comparing the output luminous intensity of the test source with that of the standard source. This involves the introduction of a third device, that of a transfer device. In our example, this device must be able to compare the luminous intensity of both sources and return values for each lamp that are proportional to the luminous intensity of the lamp, or a value

For accurate measurements, it is imperative that the transfer device be capable of comparing the exact quantity that we wish to measure. This includes both the geometrical aspects and the spectral aspects that we discussed in Section 4. The general schematic for this process is

> geometric resolution

The spectral resolution component indicated for the transfer device is not necessarily the only spectrally important component. In the case of photometric and colorimetric measurements the spectral responsivity of the complete transfer device must be equal to the desired CIE function. If this is not the case, an \* *F* will need to be determined as indicated in

If the transfer device is only used to measure the output of the test source by comparison

T S TST S S *i S SS i*

where <sup>T</sup> *S* and <sup>S</sup> *S* are the desired quantities for the test source and the standard source, and

to be applied to Equation (34) as necessary. If the standard source is a measurement

*i* are the signals from the transfer device when measuring these sources. An \* *F* is

with the standard source, the measured value of the test source is simply:

spectral resolution

Transfer device

detector

*i i* (34)

Signal

or

**5. Measurements and calibrations** 

shown in Figure 13.

Section 4.2.5 above.

T *i* and <sup>S</sup>

Source Standard (S) or Test (T)

Fig. 13. Measurement Transfer Device

that is the ratio of the luminous intensities of the two sources.

standard with accompanying uncertainties, and the uncertainties of the transfer process are accounted for, the test source is now said to be calibrated. (VIM, Section 2.39).

If the intent is to use the transfer device for subsequent measurements of the same quantity,

the factor S S *S i* may be considered a calibration factor for the transfer device, to be used to

convert the subsequent <sup>T</sup> *i* measurements into the corresponding <sup>T</sup> *S* . If the uncertainties of the standard and the process of measurement of the standard by the transfer device are considered and included, the transfer device may now be said to be calibrated for the measurement of the quantity of the standard. The calibration factor will have units of the ratio of the units of the standard source to the units of the signal. In the case of an illuminance meter calibration using the configuration of Figure 12, the units of the calibration factor for the photometer would be lux per ampere.

Figure 13 shows the transfer device as a detector unit. It is also possible that a radiation source be a transfer device. For example, if we wish to transfer a calibration between two photometers, from a calibrated standard photometer to a test photometer, we would use an incandescent lamp as the transfer device. Although it would not be necessary that this lamp be calibrated, the lamp should produce a geometric and spectral radiation field that is appropriate for the measurements to be performed by the photometers and the quantity calibration to be transferred between the photometers.

There are many measurement configurations possible that depend upon the type of measurement standard that is available and the required measurement. We will discuss several of the common configurations in the following subsections.

## **5.1 Total flux**

The measurement transfer device for total flux is the integrating sphere and detector unit shown in Figure 2, where the detector could be a spectroradiometer for spectral radiant flux measurements. The standard source and the test source are sequentially placed at the center of the sphere and the signal from their flux output is recorded at the detector. The measured/calibrated total flux for the test source is given by Equation (34), with an \* *<sup>F</sup>* applied for photometric measurements. In addition to the detector, the integrating sphere walls and all the interior components such as baffles and lamp supports must be considered as part of the responsivity *R*( ) of this transfer device. If the standard source and the test source have different output spatial distributions, any geometrical differences in the responsivity of the sphere may need to be considered. A detailed consideration of integrating sphere measurements has been given by (Ohno, 1997).
