1. Introduction

Illuminance and luminance meters, which will be indistinctly denoted as photometers in this text, are devices designed to measure optical radiation according to its action upon the CIE standard photometric observer, whose relative spectral responsivity curve is conformed to the spectral luminous efficiency function for photopic vision, V(λ), defined in ISO 23539:2005(E)/CIE S010/E:2004 [1]. The accuracy of photometric measurements is very dependent on the match between the relative spectral responsivity of photometers and V(λ).

In order to obtain V(λ) matching of photometers, detectors are usually filtered to modify their spectral responsivity to be as close as possible to V(λ). The V(λ) mismatch affects the dependence of the luminous responsivity of the photometer on the spectral power distribution (SPD) of the light sources to be characterized. In these terms, an ideal photometric performance would mean constancy of the luminous responsivity under light sources with different SPDs, and any deviation from this constancy would be considered as photometric measurement error. This error can only be corrected if the SPD of the light source is known, and it is not always the case.

For general lighting conditions –in particular, white light sources–, the Commission Internationale de l’Eclairage (CIE) recommends to evaluate the V(λ) mismatch by the General V(λ) Mismatch Index, $f_1^{{ }^{\prime }}$, which is defined as:

$f_1^{{ }^{\prime }}$ provides a general description of the photometric performance of photometers, regardless the SPD of the light source to be evaluated, but cannot be used for correction. This index was reasonably defined to quantify the spectral mismatch between the spectral responsivity of a photometer and V(λ). However, it has been observed that it is not always well-correlated with the photometric measurement error introduced by this mismatch, because the kind of SPDs of the light sources to be measured plays an important role in this error too. For instance, it was observed that $f_1^{{ }^{\prime }}$ predicts specially bad the photometric measurement errors of colored LEDs [3,4]. The reason is mainly that the V(λ) mismatch at the narrow spectral range of emission of these LEDs is unrepresentative of the V(λ) mismatch across the full spectral range. For this reason, other alternative mismatch indexes were proposed for colored LEDs with narrowband SPDs [3–5]. Some of them were based on the estimation of the photometric measurement error using a number of LED testing functions [3–5], but it was proved that a better prediction of that error is obtained by using a partial $f_1^{{ }^{\prime }}(f_{1,\text{PART}}^{{ }^{\prime }})$, which is defined similarly to $f_1^{{ }^{\prime }}$ [Eqs. (1) and (2)], but, instead of normalizing by the CIE Standard Illuminant A as in Eq. (2), a colored LED with the same dominant wavelength that the colored LED to be measured is used. Therefore, different normalizations need to be used for different colors. In this case, the problem is related with narrowband SPDs, and its origin is clear: an unrepresentative spectral mismatch at the narrow spectral range of emission of the LEDs.

We have shown above that, for the narrowband SPDs of colored LEDs (light sources not relevant for general lighting) a mismatch index could be proposed which correlates better than $f_1^{{ }^{\prime }}$ to the general photometric measurement error. But, what about other kinds of SPDs, which may be more relevant for general lighting? Is there an alternative index to $f_1^{{ }^{\prime }}$ which improves that correlation? Krüger and Blattner [6] only could relate the $f_1^{{ }^{\prime }}$ value with the maximum attainable photometric measurement error, and they found different relations for colored LEDs, phosphor-based LEDs and blackbodies. In their study, they used the Spectral Mismatch Correction Factor to account for the error, that will be explained and used in section 3. According to other studies on white LEDs, photomoters having a $f_1^{{ }^{\prime }}$ value lower than 0.03 due to the spectral mismatch will have photometric measurement error lower than 2 % [4,7]. The SPDs of white phosphor-based LEDs are composed of a narrowband and a broadband emission (fluorescence). These light sources are becoming increasingly important in lighting since the incandescent lamps are being phased out. We studied in detail the correlation between the photometric measurement error and $f_1^{{ }^{\prime }}$, and we found that it was possible to propose an alternative mismatch index not only for these LEDs, but also for other light sources with exclusively broadband emission, as it is the case of blackbodies.

We assesed the linear correlation of the $f_1^{{ }^{\prime }}$ with the general photometric measurement error for four sets of light sources: R, G and B LEDs (narrowband SPDs), blackbodies at different color temperatures (broadband SPDs), phosphor-based LEDs at different correlated color temperatures (SPDs with narrow- and broadband features), and a mixture of blackbodies and phosphor-based LED sources. In the case of the broadband SPDs, the origin of the non-correlation is other than that above described for narrowband SPDs. For broadband SPDs, it’s possible to have a spectral mismatch too small to affect the measurement, but so widely distributed across all the spectral range that highly contributes to the $f_1^{{ }^{\prime }}$ index. The photometric measurement error depends on the spectral distribution of the difference between V(λ) and the relative spectral responsivity of the photometer, and not only on the accumulated difference across the spectral range, provided by $f_1^{{ }^{\prime }}$.

The index $f_1^{{ }^{\prime }}$ is widely-used in photometry, and it is very relevant for selecting photometers for low-uncertainty photometric measurements. We think that a new index substantially better in the assessment of the general photometric measurement error of photometers will contribute in great extent to the improvement of the photometric measurements, since it allows a better selection of photometers according to their general photometric performance, and, consequently, it would improve the accuracy of the photometers in use. In this article, we propose a new mismatch index that correlates considerably better with the mismatch-related photometric measurement errors of photometers under white light sources with broadband emission.

2. Correlation between the general photometric measurement error and $f_1^{{ }^{\prime }}$

The photometric measurement error of a photometer when measuring a given light source Z of spectral power distribution S_Z(λ) can be calculated as the relative difference between the reading of the photometer and the reading it would have if its relative spectral responsivity matched exactly that of an ideal photometer, V(λ). This error, ϵ_P(Z), is defined similarly as in [3]. It corresponds to the relative difference between the luminous responsivities under Z of the ideal and the test photometer with relative spectral responsivity s_rel:

ϵ_P(Z) quantifies the photometric measurement error under just a specific light source (Z). We define here a general ϵ_P(Z) of a given photometer by considering the errors for a number of light sources with different spectral power distributions, which would represent general lighting. We propose to calculate it as the average of ϵ_P(Z) for N selected light sources:

If $f_1^{{ }^{\prime }}$ is an index for the general photometric measurement error of photometers, the values obtained by Eqs. (1) and (5) should be perfectly correlated when 〈ϵ_P〉 is calculated with a representative set of light sources for general lighting.

To study the correlation between the general photometric measurement error, 〈ϵ_P〉, and $f_1^{{ }^{\prime }}$, four sets of light sources were defined:

 

Fig. 1 (Top) RGB LED set: Spectral power distributions of the blue, red and green LEDs. (Middle) BB set: Spectral power distributions of the blackbody sources with color temperatures of 2500 K, 2700 K, 3000 K, 3500 K, 4000 K, 4500 K, 5000 K, 5700 K and 6500 K. (Bottom) Ph LED set: Spectral power distributions of the phosphor-based LEDs with CCTs of 2500 K, 2700 K, 3000 K, 3500 K, 4000 K, 4500 K, 5000 K, 5700 K and 6500 K.

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The general photometric measurement error, 〈ϵ_P〉, for 77 photometers was calculated for each one of these four sets of light sources. These photometers are illuminance meters, and belong to several European national metrology institutes. They shared the measured spectral responsivity curves within the context of the European Metrology Programme for Innovation and Research (EMPIR) project 15SIB07 PhotoLED “Future Photometry Based on Solid-State Lighting Products”.

The obtained 〈ϵ_P〉 values are plotted versus the corresponding $f_1^{{ }^{\prime }}$ values of the photometers in Fig. 2. The axes are in logarithmic scale, and the error bars expand from the minimum to the maximum ϵ_P values obtained for the different light sources included in each set.

 

Fig. 2 Relationships between $f_1^{{ }^{\prime }}$ and the general photometric measurement error, 〈ϵ_P〉, when the latter was calculated using one red, one green and one blue LEDs (upper left), 9 blackbody sources (upper right), 9 phosphor-based white LEDs (lower left), and 9 blackbody sources together with 9 phosphor-based LEDs (lower right).

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The upper left plot in Fig. 2, corresponding to the RGB LED set, shows very large general photometric measurement errors in comparison with the other sets. If expressed in percentage, these errors lie between 0.5 % and 20 %. Large photometric measurement errors are produced because the integral of the difference between ${\overline{s}}_{\text{rel}}(\lambda )$ and V(λ) weighted by the SPD of the light source [see Eq. (3)]. In the case of these light sources with narrowband SPDs as the LEDs in this set, only the difference at the spectral range of emission have a high contribution to the error, and this difference can be very large or very small, regardless the shape of the spectral responsivity across the complete range. The Pearson’s linear correlation coefficient between 〈ϵ_P〉 and $f_1^{{ }^{\prime }}$ is 0.899, and there are many cases for which the general photometric measurement error of a photometer with a smaller value of $f_1^{{ }^{\prime }}$ is larger than that of a photometer with a larger value of $f_1^{{ }^{\prime }}$, as reported in [3]. However, we must notice here that this is not only specific of the narrowband light sources, but it happens for the other sets of light sources previously defined.

It can be proved that 〈ϵ_P〉 is more similar to $f_1^{{ }^{\prime }}$ when using a set of monochromatic light sources, just replacing S_Z by Dirac deltas in Eq. (5). And it also can be proved that the photometric measurement error is larger for monochromatic light sources.

The opposite case to narrowband SPDs is represented by the BB set, which exclusively contains broadband SPDs. The general photometric measurement errors of the photometers are shown on the upper right plot, in Fig. 2. For this set, in contrast to the previous one, the general photometric measurement errors are low, ranging from 0.01% to 1.8%. The errors are comparatively much smaller because a broadband SPD allows a more uniform weighting of the difference between ${\overline{s}}_{\text{rel}}(\lambda )$ and V(λ) [Eq. (3)], and negative and positive differences are canceled out in great extent through the integration. On the other hand, the correlation coefficient is much smaller (0.772). Since the integration of absolute values at different intervals of a function [Eq. (1)] is not the same than the absolute value of the integration of that function over the full interval [Eq. (2)], the linear correlation between the values obtained by Eqs. (1) and (5) gets the worse the broader are the SPDs of the selected light sources. The reason is that, whereas differences are accumulated when calculating $f_1^{{ }^{\prime }}$ [Eqs. (1) and (2)], differences are canceled out when calculating 〈ϵ_P〉. Notice that for the BB set, for which $f_1^{{ }^{\prime }}$ should provide good predictions of the general photometric measurement errors, there are photometers with the same value of 〈ϵ_P〉, but with very different values of $f_1^{{ }^{\prime }}$ as, for instance, 0.02 and 0.07 (where 97.4 % of the values lie between 0.012 and 0.09).

The Ph LED set containing phosphor-based LEDs is an intermediate case, since these SPDs have both broadband and narrowband features (see Fig. 1). Consequently, as it is shown on the lower left plot in Fig. 2, the linear correlation is slightly larger than for the BB set case (with a correlation coefficient of 0.791), and the values of the general photometric measurement error range from 0.02 % to 2.3 %, worse than for the BB set (mainly for photometers with low values of $f_1^{{ }^{\prime }}$), and much better than for the RGB set.

A very clear improvement in the linear correlation is observed when the BB and Ph LED sets are combined in a single set (W set), from 0.772 and 0.791, respectively to 0.845 (see lower right plot of Fig. 2). This improvement is achieved because the values of 〈ϵ_P〉 of photometers with low values of $f_1^{{ }^{\prime }}$ decrease with respect to those of the Ph LED set.

3. Proposal of a new general photometric measurement error index for photometers

It may be assumed that there is a set of light sources which describes general lighting conditions, and that the general photometric measurement error can be properly quantified by 〈ϵ_P〉 just using that set into Eqs. (3) and (5). However, technological advances may vary the representative light sources for general lighting, and we think that it would be of great interest to have an index not based on specific representative SPDs, as $f_1^{{ }^{\prime }}$, that could describe accurately the general photometric measurement error under general SPDs. We propose here a new general photometric measurement error index $(f_1^{{ }^{″}})$ for broadband SPDs to be used in general lighting, which can be calculated without a well-established set of light sources. To define this general index some of the comments and results in the previous section are considered. Firstly, spectral difference compensation happens in the actual photometric measurement error, unlike the calculation of $f_1^{{ }^{\prime }}$, which is an integration of accumulated differences. Therefore, a new general index must include integration of compensated spectral differences to correlate with the real error, that is, no absolute value of these differences can be applied before integration. Secondly, a broadband light source allows a more uniform weighting of the differences between ${\overline{s}}_{\text{rel}}(\lambda )$ and V(λ) in the photometric measurement error [Eq. (3)], because negative and positive differences are canceled out in a greater extent through the integration. Within a spectral range for which the SPD is monotonic, the differences at two separate wavelengths are more likely to be compensated the closer those wavelengths are each other. As a consequence, the variation between ${\overline{s}}_{\text{rel}}(\lambda )$ and V(λ) at high spectral frequencies (defined analogously to the temporal frequency in a temporal signal) will have a lower contribution to the general index than the variation at low spectral frequencies.

We define this index $f_1^{{ }^{″}}$ from the variation of the relative spectral responsivity with respect to V(λ) once the variation at high-spectral frequencies is filtered out.

Formally, the variation of the spectral responsivity with respect to V(λ) is written as:

Then, the proposed index for the general photometric measurement error is expressed as:

With the proposed definition, the index is a low-frequency standard deviation of δ_s, unlike $f_1^{{ }^{\prime }}$, which considers the variation regardless the spectral frequency.

However, to complete this definition, the value of ν_λ,c must be determined. We have done it by studying how this value affect the correlation between $f_1^{{ }^{″}}$ and 〈ϵ_P〉. The Pearson’s linear correlation coefficients obtained for the four sets of light sources were calculated for ν_λ,c = 2 × 10⁻³ nm⁻¹, 3 × 10⁻³ nm⁻¹, 4 × 10⁻³ nm⁻¹, 5 × 10⁻³ nm⁻¹, and 6 × 10⁻³ nm⁻¹, and they are shown in Fig. 3, along with the correlation coefficients obtained for $f_1^{{ }^{\prime }}$. The figure reveals that values of ν_λ,c = 3 × 10⁻³ nm⁻¹ or 2 × 10⁻³ nm⁻¹ optimize the correlation with the sets of broadband light sources BB and W, providing correlation coefficients above 0.965, clearly in contrast with the correlation coefficient obtained for $f_1^{{ }^{\prime }}$ with these sets (0.772 and 0.845, respectively). But the correlation improvement of $f_1^{{ }^{″}}$ with respect to $f_1^{{ }^{\prime }}$ is much smaller with the Ph LED set (0.816 vs. 0.791). It is likely due to the fact that there is a narrowband emission at blue wavelengths in the SPDs (see Fig. 1). Notice that the photometric measurement error depends on the spectral difference in that narrow spectral range when it is comparable to the spectral difference across the complete spectral range. It can be proved by removing from the Ph LED set the two SPDs with a relative higher contribution of this narrowband emission (the coldest LEDs with CCT = 5700 K and 6500 K in Fig. 1), and recalculating its correlation coefficient. Then, a better improvement is obtained, with values of 0.844 and 0.746 for $f_1^{{ }^{″}}$ and $f_1^{{ }^{\prime }}$, respectively.

 

Fig. 3 Determination of the cutoff spectral frequency, ν_λ,c in the definition of $f_1^{{ }^{″}}$. The figure shows the correlation coefficients of 〈ϵ_P〉 with $f_1^{{ }^{″}}$ for the four sets of light sources when the latter were calculated for ν_λ,c = 2 × 10⁻³ nm⁻¹, 3 × 10⁻³ nm⁻¹, 4 × 10⁻³ nm⁻¹, 5 × 10⁻³ nm⁻¹, and 6 × 10⁻³ nm⁻¹. For comparison, the correlation coefficients for $f_1^{{ }^{\prime }}$ are shown in the graph too. Every position at the X-axis identifies different indexes, from the conventional $f_1^{{ }^{\prime }}$ to the indexes obtained from our analysis at different cutoff spectral frequencies.

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The RGB LED set contains just narrowband SPDs and it is not relevant for the definition of the new index. In this case, $f_1^{{ }^{\prime }}$ correlates better than $f_1^{{ }^{″}}$ with the general photometric measurement error when the latter is calculated with cutoff spectral frequencies lower than 1.4 × 10⁻² nm⁻¹, but for larger cutoff spectral frequencies both correlations are very similar (0.910 for $f_1^{{ }^{″}}$ and 0.899 for $f_1^{{ }^{\prime }}$). Interestingly, the inverse of 1.4 × 10⁻² nm⁻¹ is around 70 nm, which approximately corresponds to the average emission spectral bandwidth of the R, G, and B LEDs included in the set. It may mean that the cutoff spectral frequency to be selected to obtain an index with a good correlation with the general photometric measurement error is quite related with the representative bandwidths of the SPDs to be measured.

We decided to select a ν_λ,c = 3 × 10⁻³ nm⁻¹ instead of 2 × 10⁻³ nm⁻¹ in the $f_1^{{ }^{″}}$ definition, because the former is slightly better for the Ph LED and W sets. However, ν_λ,c values between 2 × 10⁻³ nm⁻¹ and 3 × 10⁻³ nm⁻¹ would have a similar performance.

The relation between $f_1^{{ }^{″}}$ and the general photometric measurement error, 〈ϵ_P〉, is represented in Fig. 4, analogously to the relation of $f_1^{{ }^{\prime }}$ and 〈ϵ_P〉 in Fig. 2. Both figures have to be compared in order to understand the improvement achieved by using $f_1^{{ }^{″}}$ instead $f_1^{{ }^{\prime }}$ to describe the general photometric measurement error of a photometer when using SDPs with broadband features. It is unexpected that there is a slightly worse linear correlation for the Ph LED set than for the RGB LED set, because the new index was devised for broadband SPDs. We think that the reason is that the values of 〈ϵ_P〉 are around one order of magnitude lower for the Ph LED set. It scatters more the points and that worsens the correlation.

 

Fig. 4 Relationships between $f_1^{{ }^{″}}$ and the general photometric measurement error, 〈ϵ_P〉, when the latter was calculated using one red, one green and one blue LEDs (upper left), 9 blackbody sources (upper right), 9 phosphor-based LEDs (lower left), and 9 blackbody sources together with 9 phosphor-based LEDs (lower right).

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A similar analysis can be done using, instead of the photometric measurement error ϵ_P(Z) [Eq. (3)], a quantity related with the Spectral Mismatch Correction Factor, SMCF, which is a factor widely-used in photometry. The results of both analyses are quite coherent as we will show below.

For photometric measurements, it is usual to calibrate the luminous responsivity of photometers using the CIE Standard Illuminant A. However, as explained in the introduction, this responsivity varies with the light source, and the SMCF can be applied when measuring a light source with a SPD different to that of the CIE Standard Illuminant A [2]. The SMCF is expressed as:

By definition, the lower the general photometric measurement error introduced by a photometer, the lower the expected variation of its luminous responsivity under different light sources, and, consequently, the lower spectral mismatch correction must be generally applied. The average correction required for a photometer can be quantified as the average of |1 – F^*| across a number of SPDs used both for S_Z and S_C. It is important to obtain an average across light sources, because the SMCF is affected by the spectral mismatch of both S_Z − S_C and s_rel − V, and here only the photometer performance intends to be assessed.

The relationship between the averaged |1 – F^*| across the SPDs in the W set and the values of $f_1^{{ }^{\prime }}$ (left plot) and $f_1^{{ }^{″}}$ (right plot) are shown in Fig. 5. The error bars were omitted in this case because are too large and do not provide valuable information. The index proposed in this article improves the correlation coefficient from 0.876 to 0.962, which is very similar to the improvement observed when studying the linear correlation with 〈ϵ_P〉 under the W set, as expected. When the BB set is used the correlation improvement is from 0.765 a 0.970, similar numbers to the comparison with 〈ϵ_P〉 too. These results prove that both analyses are equivalent.

 

Fig. 5 Relationships between $f_1^{{ }^{\prime }}$ (left plot) and $f_1^{{ }^{″}}$ (right plot) and the averaged |1 – F*| across the SPDs in the W set.

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We have shown that $f_1^{{ }^{″}}$ has a significantly better linear correlation than $f_1^{{ }^{\prime }}$ with respect to the general photometric measurement error. It allows a better selection of photometers according to the end-use application requirements, and, consequently, contributes to the general improvement of photometric measurements.

4. Conclusion

The linear correlation between the general photometric measurement error and the General V(λ) Mismatch Index, $f_1^{{ }^{\prime }}$, of 77 photometers has been studied for four sets of light sources: RGB LEDs (narrowband SPDs), blackbodies at different color temperatures (broadband SPDs), phosphor-based LEDs at different correlated color temperatures (SPDs with narrow- and broad-band features), and a mixture of blackbodies and phosphor-based LED sources. It has been proven that $f_1^{{ }^{\prime }}$ correlates better with the general photometric measurement error when the latter is calculated using light sources with narrowband SPDs than when calculated with broadband SPDs. Based on the observed results, a new index for the general photometric measurement error has been proposed, which involves filtering the high spectral frequencies variations between the relative spectral responsivity of the photometer and the luminous efficiency function for photopic vision, V(λ). A cutoff spectral frequency of 0.003 nm⁻¹ has been selected in the definition of the new index. At this cutoff spectral frequency, the proposed index notably improves the correlation with the real general photometric performance error of photometers for broadband SPDs, which are convenient for general lighting. For narrowband SPDs, the performance of $f_1^{{ }^{\prime }}$ is not improved by the new index. The proposed index allows the quality of the photometers to be described from just its spectral responsivity, and we consider that its use for the assessment of the general photometric performance of photometers would contribute in great extent to the general improvement of photometric measurements. Since $f_1^{{ }^{″}}$ has a significantly better linear correlation than $f_1^{{ }^{″}}$ with respect to the photometric measurement error, the selection of photometers according to the end-use application requirements can be improved, and the general photometric measurement error of photometers can be much better assessed, which is of great relevance for a good estimation of the uncertainty budget when measuring light sources for general lighting with unknown spectral power distributions.