General V(λ) mismatch index: History, current state and new ideas

The general V ( λ ) mismatch index f 1 ′ specifies the mismatch between the relative spectral responsivity of a photometer, and the spectral luminous efficiency function for photopic vision, V ( λ ). A short review of its historical development explains the reasons for the current definition and which adjustments may be helpful in the future. The properties of the current definition are described in detail. It is very likely that photometers will be calibrated with a white light-emitting diode (LED) light source as the reference in the future. This might involve the need for a more appropriate definition of the general V ( λ ) mismatch index, either by using a different normalisation in f 1 ′ for the relative spectral responsivity of the photometer or by introducing a different type of function for assessing the mismatch. On the other hand, the measurement of coloured LEDs is also becoming increasingly important. So, is a single general mismatch index for white and coloured light sources sufficient?


Motivation, specific objective
The general V(λ) mismatch index, f 1 ′ , quantifies the mismatch between the relative spectral responsivity of a photometer, s rel λ ( ), and the spectral luminous efficiency function for photopic vision, V(λ). It was defined for the first time by the CIE in 1982 1 to describe the photometric performance of photometers under general lighting conditions, and a value close to zero denotes photometers that require minimal corrections under light sources with spectral distributions (SDs) different from the defined reference sources used in photometry.

Original f 1 ′ definition
Photometer response is based on a combination of the relative spectral responsivity s rel λ ( ) and the source SD. Since incandescent light played an important role in general illumination, the photometer responsivity was first normalised with the relative SD of CIE standard illuminant A, S A (λ), to consider the mismatch (Equation 1) as defined in ISO/CIE DIS 11664. The general V(λ) mismatch index f 1 ′ specifies the mismatch between the relative spectral responsivity of a photometer, and the spectral luminous efficiency function for photopic vision, V(λ). A short review of its historical development explains the reasons for the current definition and which adjustments may be helpful in the future. The properties of the current definition are described in detail. It is very likely that photometers will be calibrated with a white light-emitting diode (LED) light source as the reference in the future. This might involve the need for a more appropriate definition of the general V(λ) mismatch index, either by using a different normalisation in f 1 ′ for the relative spectral responsivity of the photometer or by introducing a different type of function for assessing the mismatch. On the other hand, the measurement of coloured LEDs is also becoming increasingly important. So, is a single general mismatch index for white and coloured light sources sufficient?
According to the ISO/CIE 19476, 3 f 1 ′ is then defined as shown in Equation (2) The original definition of the general V(λ) mismatch index f 1 ′ for the function V λ ( ) can be generalised to any other normalised target function s T λ ( ) required to be spectrally matched.
For this purpose, a normalised spectral responsivity s rel T , * λ ( ) is calculated using the relative SD of CIE standard illuminant A, S A (λ), as a weighting (Equation (3)). The general mismatch index f 1 ′ ,T for a specific target function s T λ ( ) can then be defined as (Equation (4) The following rules for the wavelength interval of the integrals have been used in this paper: Nowadays, almost all sources to be measured in photometry are light-emitting diode (LED)-based. Therefore, in the near future, it is very likely that another standard illuminant will replace the current one (CIE standard illuminant A, S A (λ)) in the calibration of photometers. It is planned to use the SD of an LED illuminant, S L (λ), which might involve the need for a more appropriate redefinition of the general V(λ) mismatch index, either by using a different normalisation in f 1 ′ for the relative spectral responsivity of the photometer or by introducing a different type of function for assessing this mismatch. Furthermore, the present general mismatch index might not predict the expected range of spectral mismatch errors when measuring coloured LED-based light sources.

Spectral mismatch correction factors
The luminous responsivity of a photometer depends on the SD of the calibration light source, S C λ ( ). When used for measuring a light source with a different SD, S Z λ ( ) , this luminous responsivity must be corrected using the spectral mismatch correction factor (SMCF) .This factor can be expressed by Equation (5)  The integral ratio in Equation (6) results in a wavelength-independent normalisation factor a c . Therefore, the SMCF in Equation (5) can be rearranged while using Equation (6) with s as rel r el ,T,C C λ λ ( ) = ( ) to Equation (7). Equation (7) shows the connection of the SMCF in Equation (5) with the general mismatch index through the normalisation s rel, , T C λ ( ) of the relative spectral responsivity s rel λ ( ) and shows its practical importance.

History
The first notation of f 1 ′ can be found in CIE 53 1 as an informative note. The preferred measure in CIE 53 1 for quantifying the mismatch, f 1,CIE , was the maximum deviation due to spectral mismatch for five defined light sources with SDs S i Z, λ ( ) as shown in Equation (8).
Later in CIE 69, 4 the general V(λ) mismatch index f 1 ′ was introduced as a CIE recommendation. In preparation for these early CIE publications, numerous articles (e.g., Krochmann and Reissmann, 5 Krystek and Erb 6 and Krochmann and Rattunde 7 ) were published, especially in German. These articles systematically showed that the selection of f 1 ′ was reasonable under the technical conditions of that time and proved the advantages over other methods under discussion at that time.
In particular, the derivation and justification for the normalisation of the relative spectral responsivity s rel * λ ( ) using the relative SD of the calibration light source (denoted here as s rel T C , , λ ( ) in Equation (6)  ( ) in the bibliographical source) can be found in Krochmann and Reissmann. 5 The discussions can be summarised as follows: The difference between the realised and targeted functions can be defined by Equation (9): However, this type of calculation generally leads to overestimating the goodness of the fit. For the sake of comparison, it is more appropriate to show the relative error f λ ( ) instead of the difference ∆s λ ( ), as shown in Equation (10).
This is the measurement deviation for monochromatic light of wavelength λ. This can be modified to express the measurement deviation at wavelength λ when a photometric detector is calibrated using the usual calibration source, CIE standard illuminant A, by introducing s rel * λ ( ) from Equation (1) into Equation (10) instead of s rel λ ( ). Finally, this approach was extended in Geutler et al. 8 and CIE 53 1 to a more general relative error metric, denoted as f 1 ′ in CIE 53, 1 given by Equation (11).
A good summary of all the previous literature can be found in Krochmann and Rattunde. 7 It contains many calculations with different quality metrics and correlations, using a set of 223 detectors and 46 light sources (41 different others besides the five CIE light sources from CIE 53 1 ), showing the correlation between quality metrics and SMCFs.
A more theoretical approach was published in Krystek and Erb. 6 This involved a separation of the spectral mismatch error due to the photometer spectral response from that due to the spectral characteristics of the calibration source. In this approach, the detector-related dependence was described through Equation (12) and the light source-related dependence was given in Equation (13).
The definition of f Erb in Equation (12) renders the index independent of features of the calibration light source, by only including features of the detector. It was suggested as the general V(λ) mismatch index.

Current state
The current definition of the general V(λ) mismatch index can be found in ISO/CIE 19476, 3 and the use of f 1 ′ for the estimation of the maximum SMCF for phosphor-based white LEDs and RGB-type white LEDs can be found in CIE S 025:2015. 9 Here, a good correlation between f 1 ′ and the minimum and maximum SMCF of phosphor-based white LEDs was found. However, the minimum and maximum SMCF for white light produced by the mixed emission from RGB LEDs cannot be convincingly estimated based on f 1 ′ . Furthermore, the spectral mismatch error of coloured LEDs is not predictable based on f 1 ′ . This means a new or modified approach is needed for current lighting situations.

Other approaches
Czibula and Makai 10 showed that f 1 ′ is not sufficient to describe or even predict the spectral mismatch correction for LEDs, especially for coloured LEDs. The authors introduced two additional measurements using auxiliary detectors to estimate the spectral mismatch correction factor precisely.
Young et al. 11 later introduced a new quality metric f LED as the average absolute spectral mismatch error over a wavelength region relative to the error for the central wavelength of that region. Here, the influence of bandwidth changes and central wavelength changes for the spectral mismatch correction factor was discussed. The result was not a single value but a characteristic field or matrix. Csuti et al. 12 took this one step further and introduced a partial index f ′ 1,PART calculating a form of f 1 ′ for specific wavelength regions (blue, green, yellow and red) based on a calibration/normalisation with a coloured LED in that region.
A new approach for the definition of an adjusted V(λ) mismatch index f 1 ′′ was introduced by Ferrero et al., 13,14 which provides a better correlation to the average absolute spectral mismatch error than f 1 ′ , when this error is evaluated for broadband sources (phosphor-based LEDs and blackbody sources). The following sections will describe these ideas in a more detailed manner.

Czibula and Makai
Czibula and Makai 10 stated that f 1 ′ is insufficient to describe or even predict the necessary mean spectral mismatch correction for LEDs (especially coloured LEDs). They suggested an improvement of the spectral correction based on the measurement of auxiliary detectors sensitive for blue (B1) and red (R1) light, which was developed from simulations and measurements of about 1000 detectors and real/simulated spectra of about 150 LEDs.
The prediction of the SMCF using the auxiliary information was very good. There was an additional investigation to calculate the ideal pair of auxiliary detectors, but the correction results were not improved compared to the real auxiliary detectors B1 and R1 used for the first part of the work.
This is the first study that suggests that even minimal spectral information is sufficient to obtain very precise SMCF information.

Young et al.
Young et al. 11 introduced a new mismatch index called f LED (usable for coloured LEDs only), defined as the 'average spectral mismatch error over a wavelength region relative to the central wavelength of that region'. They used a model-based specific SMCF as shown in Equation (14):  (14) where S mLED (λ,• λ p ,• ∆λ• ) represents a model function for the SD of an LED with peak wavelength λ p and a full-width at half maximum (FWHM) ∆λ c . This is similar to the SMCF in Equation (5) and accounts for the spectral mismatch error of a photometer under the following conditions: • calibrated with an LED source with a peak wavelength λ λ describes the average spectral mismatch error across the wavelength range λ m ∈(p 1 , p 2 ), where p 1 and p 2 are the wavelength limits of the wavelength range considered. If we modify the FWHM parameter (calibration at FWHM ∆λ c and measuring at FWHM ∆λ m ) instead of varying the peak wavelength parameter, we obtain Equation (16).
We can define now the average spectral mismatch error while varying the bandwidth using Equation (17). Equation (17) describes the average spectral mismatch error while varying the LED model bandwidth in the range ∆λ m ∈(h 1 , h 2 ), where h 1 and h 2 are the minimum and maximum FWHM bandwidths considered. Combining the Equations (15) and (17), Young et al. 11 defined an LED mismatch index as shown in Equation (18).
According to Young et al., 11 a reasonable region for p 1 to p 2 and h 1 to h 2 would be ±5 nm around the central values of λ c and Δλ c .
This proposal by Young et al. 11 was the first to use LED models with different central wavelength and bandwidth characteristics to calculate SMCFs and summarise a specific spectral mismatch index from all this information.

Csuti et al.
Csuti and Kránicz 15 and the additional contributions from Csuti et al. 12,16 summarised discussions of CIE TC2-45 to prepare the CIE report CIE 127 17 at the end of 2003. The discussion in technical committee TC 2-45 included two key proposals. One proposal is shown in Equations (19) and (20). Note that compared with Equation (5), the weighting of the relative spectral responsivities with the relative SD of the calibration source in the first integral ratio was left out here.  The second proposal is shown in Equation (21), but this is not very helpful because problems with the spectral matching can be averaged out. However, neither of these suggestions were published in the final report CIE 127, 17 meaning that the references in Csuti et al. 12 are the only remaining information. For this definition, it is necessary to define the wavelength range (400-650 nm) and the steps (10 nm) to get consistent results. However, a definition of this form leads to problems when calculating the measurement uncertainty.
The suggestion from Csuti et al. 15 was to use four different wavelength ranges, instead of the maximum over the complete wavelength range, which was used in Equation (23) For the integration limits in the nominator see Table 1. However, the problem of Equation (25) is the lack of a reference source for each required calibration condition.
The complete calculation process is demonstrated in the following figures. Figure 1 shows the relative spectral responsivity of sample photometer #2 and V λ ( ). The normalised SDs of the reference LEDs (normalised to an integral of 1, see Table 1 for the definition of the wavelength ranges and reference LEDs) are shown in Figure 2. Using these data, Figure 3 shows the calculated differences between the normalised spectral responsivities of photometer #2 (Equation (24)) and V λ ( ) .
Further explanations for handling SMCFs, especially with monochromatic light and LED models, can be found in Sections 5.2 and 5.3.

Ferrero et al.
Ferrero et al. 13 proposed an alternative index f 1 ′′ , based on the idea that the spectral shape of the relative deviation between the relative spectral responsivity and V(λ) impacts the resulting mean spectral mismatch error. For instance, any deviations that vary rapidly with wavelength have a lower impact when broadband light sources are to be measured with a photometer because those variations are smoothed out by integration when calculating the luminous responsivity. Therefore, a formalism based on the Fourier transform of the relative error f 1 ′′ equals the standard deviation of δ s when the cut-off frequency ν λ,c is infinite, but it is lower for a finite ν λ,c . A general value of ν λ,c can be defined for general lighting purposes, but others can be recommended for more specific lighting scenarios, such as those that consist exclusively of phosphorbased LED sources. Ferrero et al. 13 showed that this alternative index results in a better correlation than f 1 ′ with the mean mismatch errors in broadband or mixed scenarios, but not for narrowband colour LEDs, where the correlation is lower.

Coloured LED modelling
A simple model S mLED λ λ λ , , 0 ∆ ( ) (Equations  ( , ,  ) , The numerator in Equation (29) was also used in Young et al. 11 for modelling coloured LEDs. More complex models could also be used, but there is no significant improvement for general purposes. However, for the specific determination of an actual SMCF, the measured SD of the LED should be taken into account and not the modelled one.
As an example of a more complex model, Ohno 18 published the first realistic model as shown in Equations (31) and (32). Note: In this model, the FWHM is about 0.9· ∆λ 0 5 , .

Spectral mismatch correction factors for monochromatic light sources
For monochromatic light sources emitting light only at a specific wavelength λ , the SMCF in Equation (7) becomes Equation (33).

Spectral mismatch correction factors for LEDs
Similarly, for LED sources, the SMCF can be expressed, from Equation (7), as shown in Equation (34):

Conclusion
This article explains the basics of the general V(λ) mismatch index f 1 ′ and its background. Furthermore, based on the published literature, it is shown that several approaches lead to better predictions of the general V λ ( ) mismatch of photometers for general lighting situations using LED sources.
The following objectives can be used as a guide for the next steps towards a new or extended definition for the quality metric used to describe the spectral mismatch of photometers: • Define a V λ ( ) mismatch index such that its values provide better correlation than the current f 1 ′ values with the average or maximum spectral mismatch error introduced when measuring light sources that are spectrally different to the calibration light source.
• The definition should be independent of any specific set of light sources. • The definition should be independent of the wavelength resolution of the relative spectral responsivity s rel λ ( ) (some minimal requirements for the sampling and bandwidth must be defined). • Take into account the requirements for simple measurement uncertainty calculation (be careful with maximum values, absolute values, etc.). • The mismatch index should be easy to implement by manufacturers and easily reproducible by users.
As an initial suggestion, we propose the following: • Change the current definition of f 1 ′ using Equations (3) and (4) to a similar one using a normalisation function for the relative spectral responsivity s rel λ ( ) of the detector, which is connected to the relative SD of the light source actually used for calibration rather than standard illuminant A, that is, replace S A λ ( ) with S C λ ( ).
• Add a different approach to f 1 ′ . Since f 1 ′ was defined only for general lighting, it might be necessary to provide a new index to evaluate the mismatch of photometers for lighting conditions under coloured LED sources. A graph should be provided showing the SMCFs for a set of simulated coloured LEDs of 20 nm FWHM and a simple model as shown in Equation (29).