High-Quality Multi-Emitter LED-Based Retrofits for Incandescent Photometric A Illuminant Reliability of R² Evaluation

Abstract

This research deals with the design problems of LED-based spectrally tunable light sources (LSTLSs). The study aims to assess the reliability of popular models for the spectral modeling of LED radiation and a typically used curve-fitting criterion (R²) in the development of high-quality multi-emitter LED retrofits for incandescent photometric illuminant. The research methodology involves modeling each LED channel using Lorentz and Gaussian functions and combining multiple channels to approximate the desired spectral power distribution (SPD). After the optimization, 20 various LED sets were designed, which allowed us to replicate the SPD of CIE illuminant A with a very high R² value. Two sets were constructed and measured to recognize the reliability of the simulation approach. The results suggest that for planning the LSTLS for photometric applications, these models are unreliable as they do not reflect the real effect of changes in the characteristics of the components nor reveal the share of various spectral ranges. Therefore, the decisions made on these criteria may not be the best solutions in the context of specific applications.

1. Introduction

Accurate photometric measurements require the calibration of photometric meters. According to [1], for this procedure, a light source characterized by the spectrum of an CIE illuminant A is required, which is a blackbody radiator at a temperature of 2856 K. Incandescent illuminants provide predictable spectral output, stable if correctly supplied, but are inefficient and have limited spectral capabilities [2]. The rapid advancement of LEDs in optical power and wavelength range makes LED-based spectrally tunable light sources (LSTLSs) promising for replicating an approximate illuminant A spectrum [3], addressing challenges from the discontinuation of incandescent standard lamps.

Recently, multi-emitter LED systems have become a compelling retrofit solution for traditional light sources, offering enhanced performance, energy efficiency, and, most of all, spectral tunability with stable and adjustable light levels [4,5,6]. A unique feature of LED technology is the modulation of their emission by forward current [7] and junction temperature [8]. Careful thermal consideration is required regardless of how it is perceived—as an advantage or disadvantage.

Researchers have explored LSTLSs as a potential calibration alternative since the early 2000s [9,10,11,12]. The main benefit of an LSTLS is its ability to generate different spectral power distributions (SPDs) and illuminances with a single system, eliminating the need to exchange the light source or add optical filters [13,14]. Initially, LSTLs utilized multiple LEDs within specific wavelength bands, primarily limited to the visible range [15]. Further, the advancements in LED technology have enabled high-power LEDs to emit sufficient optical power across broader wavelengths, including ultraviolet and infrared regions, which is especially useful for testing photovoltaic modules [16].

The goal of research on LSTLSs has always been to accurately reproduce various SPDs, particularly CIE standard illuminants, with high flexibility and luminous efficacy [6,15]. Current research efforts primarily focus on increasing the number of channels (wavelengths of LEDs) and the number of emitters in each channel to extend emission ranges and enhance the precision and speed of spectral matching through advanced algorithms [15].

Although several papers have focused on the spectral matching of CIE illuminants for various applications, for example, calibration of illuminance meters [9], colorimetry [11,12,13,15], testing photovoltaic modules [16,17], calibration of optical sensors [18], general lighting [19,20,21], vision research [22], textile industry [23], or medical illumination and imaging [6,24,25], only a few have addressed selecting the optimal set of LEDs for simulations [17,26,27]. This is crucial because, with a few dozen LEDs available from manufacturers, there are many possible combinations for adjusting their performance.

The National Institute of Standards and Technology (NIST) and Philips [27] have developed an LSTLS luminaire for general lighting that utilizes 22 different LED wavelengths (channels) and provides nearly 100,000 lumens, generating a comprehensive range of CIE daylight sources and other SPDs. Fryc et al. [13] proposed a 35-LED channel LSTLS and integrating sphere (30 cm diameter). They used a spectral matching algorithm with an iterative convergence procedure based on partial derivatives. To evaluate the quality of the obtained SPD, the ratio of the integrated absolute difference between test and target source distributions to the integrated target source distribution (standard illuminants) was applied. Mackiewicz et al. [22] used a minimization algorithm to avoid lengthy convergence procedures, allowing each LED’s linear and repeatable luminance adjustment. Their system included a one-meter integrating sphere and six spectrally programmable light sources, though the number of spectral channels was insufficient for precise matches to real illuminants. The purpose of the system related to vision research, and therefore, the Color Rendering Index (CRI) was used to analyze the differences between spectra. Burgos-Fern et al. [12] integrated 31 different LEDs (from 400 to 700 nm) into a cube capable of producing D65, D50, A, and E illuminants and mitigating the SPD of fluorescent and high-pressure lamps. They used a large number of low-power LEDs, and therefore, the spectral shift could be neglected. Apart from colorimetric criteria (correlated color temperature CCT, color rendering indexes Ra and Rb, chromaticity coordinates CIE-xy), to evaluate the differences between intended and obtained spectra, they used the goodness-of-fit coefficient (GFC), mean absolute error (MAE), and root mean square error (RMSE). Colorimetric criteria are not fully reliable for evaluating the fit quality due to the metamerism effect [2]. Godo et al. [9] developed an illuminance meter calibration system utilizing the LSTLSs to replace traditional incandescent lamps. Their source comprised 23 types of LEDs, including monochromatic and white high-power LEDs operating in the wavelength range from 340 nm to 800 nm. It reproduced the A illuminant A within illuminance levels from 800 lx to 10,000 lx, achieving a coincidence error (ΔES-A) below 7.5%. Recently, Godo and Watari [10] used the developed LSTLS to additionally simulate the illuminant D50 for imaging and graphics fields. The quality of the D50 simulator was evaluated by applying Ra and Δu’₁₀v’₁₀ parameters. The LSTLS developed by Llenas and Carreras [21] has 10 chromatic LED channels spread across the visible spectrum. They investigated three different heuristic approaches: (i) a genetic algorithm, (ii) simulated annealing, and (iii) a Monte Carlo simulation. The error functions used were the mean absolute percentage deviation (MAPD), measured for each wavelength (to estimate the quality of spectral matching), and the color difference Δu’v’ with assumed target spectra. None of these investigations included the effect of spectral components’ selection. However, other researchers have noticed this aspect.

Hu et al. [26] developed an algorithm for determining the optimal radiant flux for each LED in the LSTLS to synthesize spectra by maximizing averaged LED utilization rates. They revealed that peak wavelength, full-width half maximum (FWHM), and radiant flux are crucial for optimal LED selection. Later, the optimal selection of LEDs for realizing CIE standard illuminants and blackbody radiation at different color temperatures was analyzed by Wu et al. [28]. They proved that careful selection of peak wavelengths and FWHM values allows for minimizing the mean square error (MSE) of the synthesized SPDs. Lukovic et al. [18] proposed a novel numerical method for evaluating the contributions of each LED channel in LSTLSs. Due to their intended application (the calibration of various optical sensors), their selection criterion was applied in the spectral range from 700 nm to 1070 nm. They noticed that the SPD of each LED could be initially approximated by a Gaussian curve to accelerate calculations. Still, they suggested that final calculations should be performed with real (measured) SPDs.

Recently, researchers have investigated LSTLS by employing systematic prediction algorithms. Yuan et al. [29] utilized Gaussian and asymmetric Gaussian distribution functions to tune 32 LEDs, successfully simulating CIE A and D65 with less than a 5% deviation. Fan et al. [30] used machine learning algorithms to forecast the SPD of full-spectrum white LEDs. Meanwhile, Yuan et al. [31] employed deep learning techniques to predict the SPD of LED systems undergoing multiple degradation mechanisms. Chen et al. [32] applied differential evolution algorithms for multi-type LED-based AM1.5G solar spectrum synthesis. The spectral match was determined as the ratio of the actual percentage of irradiance that falls on the spectral range of concern to the required percentage of irradiance. Nahavandi et al. [20] studied three algorithms: equal spacing of wavelength range (“Equal”), Gram Schmidt orthogonalization in LEDs/light primaries spectral subspace (“Gram”), and a generalized version using 200 imaginary light primaries and 40 real LEDs (“Ortho”). The performance of the algorithms improved with the higher number of selected LEDs/light primaries, evaluated in terms of RMSE, GFC (goodness-of-fit coefficient—the cosine of the vector angle between the target and the reconstructed vector), ΔCCT, and CRI. The result suggested that the “Ortho” algorithm might be the most effective. However, it does not include the commercial availability of LEDs. Kumar et al. [33] presented a methodology for multi-objective optimization to choose the optimal LED set for simulating daylight illuminants: D50, D65, and D75 with CRI > 90 and an average illuminance of 2000 lx. They used a multi-objective algorithmic approach to select from 5 to 74 LED emitters.

Despite the widely understood benefits of LSTLSs and their application potential, a notable gap in research regarding their photometric quality and application in light meter calibration stands has been recognized [10]. Additionally, from the above literature review, it can be inferred that the measures used to evaluate the quality of LSTLS spectral matching are usually based on the criteria that average the differences in the spectra. Additionally, many different solutions are characterized by very high (similar) values of the quality parameters, which could suggest the ineffectiveness of some of these constructions understood as the use of an overestimated number of channels. Our research aims to critically evaluate traditional modeling techniques against measurements to determine their reliability in designing LED replacements for incandescent sources in photometric applications, specifically for photometric calibration procedures. We explored the use of Gaussian models and an exemplary evaluation method (R-squared curve fitting—R²), widely used to simulate the SPDs of modern LSTLSs based on technical data from LED datasheets, and assessed their fidelity against empirical results. By comparing the effectiveness of these models with exemplary measurements, we aim to identify the areas for improvements in the development of LSTLSs. The main goal is to establish the reliability of these techniques in designing and validating photometric illuminants, providing a systematic approach to retrofitting incandescent sources with advanced LED technology. The findings from this study are intended to refine the understanding and improve the design processes of LSTLSs for photometric applications.

2. Materials and Methods

The research involved four main steps: 1—spectral modeling of LED channels, 2—selection of LEDs for the LSTLS, 3—channels’ optimization, and 4—evaluation of the reliability of the investigated models and measures. In this study, we included one target SPD of the LSTLS (an A illuminant A SPD) as it is used to calibrate photometric instruments.

2.1. Spectral Modeling of LED Channels

The LED emitters are characterized by multiple parameters, specifically the peak wavelength λ_p, FWHM, and optical power P_opt. In this research, the availability of components with various λ_p was monitored. In some cases, emitters with the same λ_p but various FWHMs may be available. Thus, at the beginning of the research, the manufacturers’ data of high-power LEDs were used to design the SPDs mimicking the illuminant A. The spectral radiant power P(λ) of the chromatic channel was simulated with two models: the Gaussian model (Equation (1)) and the Lorentz model (Equation (2)) [34]:

$${SPD}_G=P\left(\lambda \right)=P_{max}·\frac{exp\left(\frac{-4ln\left(2\right){\left(\lambda -{\lambda }_p\right)}^2}{{FWHM}^2}\right)}{FWHM·\sqrt{\frac{\pi }{4ln\left(2\right)}}},$$

(1)

$${SPD}_L=P\left(\lambda \right)=P_{max}·\frac{2}{\pi }·\frac{FWHM}{{4\left(\lambda -{\lambda }_p\right)}^2{+FWHM}^2},$$

(2)

where P_max is the maximum radiant density, λ is the wavelength, λ_p is the peak wavelength in the LED spectrum for which the radiant power is the maximum, and FWHM is the full width at half maximum in nm. More advanced functions approximating the spectral distribution of LEDs [34] require prior spectroradiometric measurements [32]. At this stage of the research, it is assumed to rely only on the manufacturers’ data.

2.2. Selection of LED Sets for the Analysis

High-power LED emitters were selected from a wide range of available spectral types, with consideration given to their peak emission wavelengths and FWHMs. Due to the lack of emitters in some spectral ranges, a white-light-emitting diode is assumed to be used. The data used during simulations for this channel were the real (measured) SPDs due to the lack of models of the radiation of this type of LED. All calculations were performed assuming the luminous flux of the reference illuminant equal to 1000 lumens. This model allows for adjusting the optical power of each channel with a defined peak wavelength and the FWHM, enabling precise control over the emitted spectrum. The optimization process involves an iterative approach where the SPD of each emitter is adjusted based on LEDs’ peak wavelength availability and the desired photometric properties of LSTLS.

The availability of LED types significantly influences the feasibility of the design, requiring a balance between the theoretical model and practical component availability. Manufacturers sort LEDs into groups based on their characteristics, ensuring complete SPD. To cover the visible spectrum ranging from 380 nm to 780 nm, LED emitters with different FWHMs (from 9 nm to 100 nm) with a step of 5 nm were selected (

Table 1. Basic data of the SMD LEDs selected for analysis: λ_p wavelength peak and FWHM (manufacturers’ data).

λp [nm]	FWHM [nm]	λp [nm]	FWHM [nm]	λp [nm]	FWHM [nm]	λp [nm]	FWHM [nm]
380	9	480	20	580	40	680	20
385	11	485	20	585	20	685	26
390	12	490	28	590	14	690	25
395	10	495	25	595	65	695	90
400	12	500	22	600	90	700	20
405	12	505	30	605	85	705	33
410	12	510	29	610	85	710	25
415	13	515	35	615	20	715	22
420	18	520	30	620	15	720	20
425	15	525	29	625	16	725	36
430	15	530	30	630	15	730	30
435	25	535	35	635	20	735	40
440	25	540	100	640	90	740	22
445	18	545	110	645	16	745	25
450	22	550	38	650	90	750	28
455	25	555	110	655	17	755	24
460	25	560	40	660	17	760	25
465	25	565	120	665	20	765	29
470	20	570	44	670	20	770	25
475	21	575	11	675	20	775	27

). Most of them are characterized by a narrow emission spectrum (FWHM = 9–40 nm). However, a lack of high-power narrowband LEDs can be observed on the market for wavelengths 540, 545, 555, 565, 595, 600, 605, and 610 nm (green, yellow, and orange light). Therefore, they were replaced by broadband PC (phosphor-converted) LEDs with FWHMs ranging from 65 nm to 120 nm.

2.3. LED Channels’ Optimization

The target spectrum was modeled using various composition versions in the third step. The SPDs of LSTLSs were generated using the Levenberg–Marquardt algorithm LMA, also known as the damped least-squares (DLS) method. This algorithm provided the combined output of multiple LED emitters. Then, the operational parameters of each channel were determined. The simulations aimed to predict the performance of the LED array before physical assembly, providing a basis for subsequent empirical validation.

The calculations of the target SPD included sets containing 21 to 83 channels, each with a different peak wavelength. Each set consisted of various numbers of chromatic channels and FWHM values (

Table 2. Components of tested versions of LSTLSs.

Version No.	Number of Chromatic Channels	White LED	Key Differences between Versions
			Step 5 nm	Step 10 nm	Step 20 nm	Narrowband LEDs 640 nm and 650 nm	Wideband LEDs 640 nm and 650 nm
1	82	No	YES				YES
2	82	No	YES			YES
3	43	No		YES			YES
4	43	No		YES		YES
5	21	No			YES		YES
6	21	No			YES	YES
7	25	No	Set of 25 selected LEDs (simulated SPDs of each channel)
8	25	No	Set of 25 selected LEDs (measurements)
9	82	2700 K	YES				YES
10	82	2700 K	YES			YES
11	43	2700 K		YES			YES
12	43	2700 K		YES		YES
13	21	2700 K			YES		YES
14	21	2700 K			YES	YES
15	82	3000 K	YES				YES
16	82	3000 K	YES			YES
17	43	3000 K		YES			YES
18	43	3000 K		YES		YES
19	21	3000 K			YES		YES
20	21	3000 K			YES	YES
21	25	3000 K	Set of 26 selected LEDs (simulated SPDs of each channel)
22	25	3000 K	Set of 26 selected LEDs (measurements)

) with or without additional high CRI (equal to 98) white-light-emitting diode of Sunlike type at 2 various CCTs. It was used to reduce the number of channels and cover the valleys in the output SPD. The research procedure was designed to verify the usability of broadband PC (phosphor-converted) LEDs. This test was performed by replacing narrowband LEDs with peak wavelengths at 640 nm and 650 nm (FWHM = 20 nm) with broadband PC LEDs with peak wavelengths at 640 nm and 650 nm (FWHM = 90 nm). In practice, the feedback control and power supply of a system with a number of channels exceeding 40 present significant challenges. Moreover, such LSTLS would be impractical in photometric layouts, such as with an integrating sphere of limited dimensions. For this research, an additional assumption related to the application was introduced, which limited the area of the LSTLS and excluded the versions with the highest number of channels for practical verification. Therefore, 2 additional sets were selected, among other results, composed of 25 chromatic channels with or without additional Sunlike white-light-emitting diodes, which met the criteria of required matching quality and the surface of the LSTLS.

2.4. Evaluation of the Investigated Models

The quality of spectral matching was assessed through the most frequently used R-squared (R²) and root mean square error (RMSE) values, which quantify the goodness of fit between the simulation data and the target [35]:

$$R^2=1-\frac{{\sum }_{\lambda =380}^{780}{\left(A_{\lambda }-M\right)}^2}{{\sum }_{\lambda =380}^{780}{\left(P_{\lambda }-M\right)}^2},$$

(3)

where P_λ—spectral power density of the source for the wavelength λ, A_λ—spectral power density of the target SPD for the wavelength λ, and M—mean value of the spectral power density for the target SPD.

$$RMSE=\sqrt{\frac{{\sum }_{\lambda =380}^{780}{\left(A_{\lambda }-P_{\lambda }\right)}^2}{N}},$$

(4)

where P_λ—spectral power density of the source for the wavelength λ, A_λ—spectral power density of the target SPD for the wavelength λ, and N = the number of data points.

Equations (3) and (4) are used to compare the spectral deviation between the modeled (in stage 1) and measured SPDs of LEDs and between the designed LSTLS and the illuminant A at the third stage of this research.

The experimental validation of the simulated LED systems was conducted using a well-calibrated photometric setup. This included a spectroradiometer GL Spectis 1.0 Touch with an integrating sphere GL OPTI SPHERE 205 mm (GL Opic, Puszczykowo, Poland) to independently measure the SPD, optical power, and luminous flux of each LED emitter. These data were used to evaluate the accuracy of spectral modeling of LEDs selected for practical tests and the quality of predicting the SPD of the LSTLS. The setup was designed to replicate standard photometric testing environments with controlled ambient conditions to minimize external influences. Each LED was mounted on a metal-core printed circuit board (MCPCB) and then on an oversized heat sink. The ambient and heat sink temperatures were constant (20 °C).

3. Results

The SPD and optical power measurement results of each LED were cross-referenced with manufacturers’ data to validate whether the LED’s behavior aligned with expectations. This validation formed the basis for preliminary analyses of the spectral distribution of the LSTLS and optimization of its SPD. The manufacturers’ data of each LED were approximated using Gaussian (Equation (1)) and Lorentz (Equation (2)) functions and compared against measured SPD. Both models provided very suitable values of measure of fit for the SPDs for each LED. The average R² value for all LEDs did not fall below 0.98 (

Table 3. Coefficients of determination (R²) for spectral power distributions of LEDs fitted with Gaussian and Lorentz functions.

		R2
Wavelength	FWHM	Lorentz	Gaussian
[nm]	[nm]	-	-
385	12	0.988	0.991
395	12	0.992	0.993
405	14	0.992	0.993
420	13	0.991	0.991
435	17	0.991	0.990
450	20	0.994	0.990
470	20	0.993	0.986
490	26	0.987	0.985
515	35	0.981	0.983
540	120	0.928	0.942
550	35	0.982	0.984
575	90	0.957	0.969
590	90	0.963	0.973
590	14	0.994	0.994
615	70	0.971	0.981
640	90	0.958	0.968
660	90	0.964	0.968
660	17	0.985	0.984
680	25	0.974	0.975
690	25	0.982	0.980
700	20	0.984	0.985
720	23	0.991	0.988
740	22	0.986	0.985
760	25	0.990	0.981
780	24	0.980	0.980
Mean		0.980	0.982

).

From Table 3, it can be inferred that for LEDs emitting some wavelengths, the fitting is, in general, worse no matter which model is applied. This occurs for some wideband LEDs (540, 575, 615, and 640 nm) and selected narrowband LEDs (590 and 680 nm). The Gaussian function exceeds the Lorentz function in reproducing steep slopes. In contrast, the Lorentz function is more accurate for more gentle slopes (see examples in Figure 1). Additionally, the Gaussian function better models the SPDs with higher asymmetry. For example, for the LED with a peak at 460 nm and FWHM = 25 nm (see Table 1 and Figure 1a), R² values were 0.942 (Gaussian) and 0.928 (Lorentz). Figure 1c presents the SPDs for the LED with the peak at 640 nm peak, for which the R² values are 0.968 and 0.958, respectively.

Considering the slightly better performance of the Gaussian model, specifically for asymmetric SPDs, it was chosen for the simulations of the LSTLS versions. The selection of LED sets with the criterion of R² above 0.99 resulted in a number of solutions for each set. In

Table 4. Key differences between simulated versions of LSTLS-R², RMSE coefficients, and the number of channels used (no. 8 and 22 are based on measurement data).

Number of the Version	R2	RMSE (mW/nm)	Number of LED Channels
			Available	Used
1	0.99999	0.026	82	72
2	0.99999	0.033	82	69
3	0.99985	0.123	43	37
4	0.99994	0.075	43	39
5	0.99707	0.543	21	19
6	0.99342	0.816	21	19
7	0.99910	0.300	25	22
8	Not included in this part of the analysis
9	1.00000	0.0029	83	68
10	0.99998	0.041	83	70
11	0.99970	0.174	44	35
12	0.99999	0.035	44	38
13	0.99479	0.812	22	20
14	0.99711	0.538	22	22
15	0.99998	0.040	83	63
16	0.99998	0.041	83	68
17	0.99971	0.170	44	35
18	0.99997	0.057	44	36
19	0.99729	0.523	22	18
20	0.99723	0.529	22	20
21	0.99887	0.336	26	22
22	Not included in this part of the analysis

, the data of the best sets are collected: the R² value, the number of channels used during optimization of each set, and the number of channels used in a final solution of each set. From the results, it can be inferred that the design of the LSTLS is a non-trivial problem. The A illuminant A LSTLS with a very suitable match of SPD evaluated with the criterion of R² value may be realized with sets composed of various numbers of channels, where each channel may be composed of various numbers of LEDs of the same type. All SPDs are characterized by a CCT of 2856 K and an R² value above 0.993. The majority (90%) of the SPDs have an R² value over 0.997, which suggests an almost perfect fit of the spectra. The analysis of the LSTLS results shows that employing 82 LEDs in each version provides the highest R² value.

Figure 2 presents the SPDs of modeled LSTLS versions (numbers according to Table 4).

In Figure 2b, the spectral power distributions of the modeled LSTLS normalized to the SPD of the illuminant A for 1000 lm of the luminous flux are presented, while in Figure 3, the spectral range is limited to 430 to 690 nm. In this spectral range, the human eye’s luminous efficacy exceeds 0.001 of its maximum value, and therefore, this spectral range is recognized as the most significant in photometric applications.

In Figure 4, SPDs are compared in pairs with only one or two differences between their components. For example, SPDs in Figure 4a differ only in terms of using colored LEDs at wavelengths 640 and 650 nm. This does not mean that part of the spectrum is subtracted but that the SPD was optimized for the difference between these two components. As both spectra were optimized for those components, the operation points of these two and other LEDs are different, and differences occur in the shape of the optimized SPD. In Figure 4d–f, an additional influence of the change in the CCT of the white-light-emitting diode is presented.

Using 43 LEDs (including broadband 640 and 650 nm) causes a slight deviation from the reference SPD within the 575–675 nm range (R² = 0.99985), while utilizing narrowband 640 and 650 nm LEDs (R² = 0.99994) mitigates this deviation, occurring only between 575 and 625 nm (curves 5 and 6 in Figure 2c and Figure 3a). Employing 21 LEDs (including narrowband LED 640 nm) results in a deviation from the reference distribution within the ranges of 380–430 nm and 530–780 nm (R² = 0.99342). Instead, employing broadband LED 640 nm (R² = 0.99707) reduces the deviation within the spectral range of 530–675 nm (Figure 4c), which holds significant importance for photometric applications.

The analysis of the version no. 9–20 indicates that utilizing 82 LEDs in the source allows reproducing the SPD of illuminant A with R² ≥ 0.99998 (Figure 2b). However, using 43 LEDs (including broadband 640 and 650 nm) causes a deviation from the reference distribution within the range of 575–675 nm (R² = 0.9997), whereas employing narrowband 640 and 650 nm (R² ≥ 0.99997) reduces this deviation, which is in contrast to an LSTLS lacking white LEDs. Employing 21 LEDs (including broadband LED 640 nm) results in a significant deviation from the reference distribution within the ranges of 380–430 nm and 530–780 nm (R² = 0.995). On the other hand, using narrowband LED 640 nm (R² ≥ 0.997) reduces the deviation from the target distribution within the relevant spectral range of 475–600 nm, which is contrary to an LSTLS lacking white LEDs.

Experimental Validation of the LSTLS Models

To validate the elaborated models, two of them, no. 7 (0.999) and 21 (0.999), were constructed and characterized (see methodology in Section 2.4). The comparison of the SPDs is presented in Figure 5.

The results suggest that including a white LED reduces the deviation from the standard spectral power distribution (SPD) for wavelengths shorter than 500 nm (Figure 5b). However, the fit is worse within the spectral range of 500–650 nm, which is extremely relevant in photometry. This is opposite to the LSTLS without a white component.

4. Discussion

The measured spectra for the designed supply currents have worse performance according to the R² measure—it is 0.992 (RMSE = 0.891 mW/nm) for SPD no. 8 and 0.991 (RMSE = 0.951 mW/nm) for SPD no. 22, which is caused by the changes in the SPD of the components for various operation points. During the characterization of LEDs, variations in their spectral parameters were observed (λ_p and FWHM) in response to changes in current intensity. The spectral shifts are not similar for all types of LEDs—the most significant shifts are observed for two spectral ranges (435–550 nm and 660–780 nm). These shifts are notably more pronounced for blue and green light compared to red light. Moreover, the shift occurs toward shorter wavelengths for LEDs emitting blue and green light, whereas for LEDs emitting red light, it occurs toward longer wavelengths. Additionally, the test results revealed a non-uniform effect of power supply on the changes in FWHM. These effects are the main reason for the discrepancies between simulated and measured SPDs, but a detailed analysis of this effect is not the merit of this research. The accuracy of the simulations of the LSTLS could be improved if the models of changes for each LED were included during simulations. This is the main limitation of this research. However, its aim was not to model accurately the LSTLS but to analyze the most widely used criterion for the quality of spectral matching.

The R² measure is widely applied for curve-fitting purposes. We applied it to evaluate the components and the system. Additionally, we calculated the RMSE values as another popular quality measure. It provides an average discrepancy of the final SPD per nanometer; therefore, both measures are similar in that both provide averaged data. They also lead to similar conclusions, as higher R² usually means lower RMSE with very small discrepancies from this rule (see Table 4). That makes it difficult to assess the quality of the spectral matching of LSTLSs, which depends on the selection of LED channels and their operation conditions, as they affect the SPD of each component of LSTLSs, even if the temperature is constant. As presented in this research, similar mismatch error values may be obtained for different SPDs where the mismatch occurs in various spectral bands. That may be observed, for example, in Figure 6, which presents the SPDs of two measured LSTLS solutions, which are characterized by similar values of R². However, the discrepancies between the target spectrum and the obtained ones occur in various spectral ranges—the central region (red circle) overlaps with the region, where the V(λ) curve has the highest values. In contrast, the border region (green circle) is less important from the point of view of photometry.

Similar doubts can be drawn for the evaluation of components used in the LSTLS for photometric applications. For example, for LEDs presented in Figure 1a,c, the values of R² were noticeably lower for LEDs with a peak at 460 nm while higher in the case of 640 nm. In spite of that, comparing their spectra, it can be inferred that in the first case, the peak and FWHM were accurately modeled. The biggest differences occurred outside the central part of the SPD. The opposite situation was for the LED with the peak at 640 nm; although the region with the highest share of the optical power density was not very well simulated, the R² value was higher. These effects suggest that it might be useful to search for other measures for the evaluation of spectral matching of SPDs of the components and LSTLS systems in specific applications, which could improve the accuracy of their modeling and facilitate the selection of the best solution. Specifically, responsivity curves adequate for the application area could be considered design criteria for the LSTLS [36].

5. Conclusions

In this research, we presented the results of simulations and validation of the LSTLS system with the R² measure. Analysis of the spectral output for different LED configurations revealed certain wavelengths significantly contributing to reducing the overall spectral mismatch, mainly due to low FWHM, in the violet-blue light region < 500 nm and deep red region > 700 nm. When there is a wider spacing between peaks, indicating fewer LED channels in the LED source, color LEDs emitting a narrow spectral spectrum should be substituted with phosphor-based LEDs emitting a broader spectral spectrum. For instance, replacing narrowband LEDs at 640 nm and 650 nm with broadband sources smoothed the final SPD curve in the 600–700 nm range, optimizing the overall spectral output. However, this adjustment for more channels in LSTLS had minimal impact on the R² value, suggesting the robustness of the selected LED configurations against minor spectral shifts.

The results of our study also suggest the benefits of integrating phosphor-converted LEDs and adjusting the bandwidth of narrowband LEDs to enhance the overall spectral output. These strategies may contribute to smoother spectral profiles and reduced overall mismatch errors, demonstrating the potential for hybrid LED configurations in critical photometric applications but providing new technological challenges. The optical properties of phosphor-based optical sources derive from several factors, such as the synthesis methods of nanoparticles embedded within the host matrix. Ion implantation as a synthesis technique allows for the precise control of nanoparticle size, distribution, and density within, which in turn affects the photoluminescence and nonlinear optical properties [37].

Our study aimed to evaluate the efficacy of Gaussian modeling and R-squared fitting criteria in designing and simulating multi-emitter LED systems, which has provided critical insights and actionable recommendations. While these traditional methodologies offer a solid foundation for the initial design phases of LSTLSs, they require significant adaptations to fully accommodate LEDs’ unique spectral characteristics and performance variabilities.

The comparison of the spectral output of optimized versions of the LSTLS indicated high fidelity between simulated and actual SPDs, with R² values exceeding 0.993 for most sets. However, comparing the errors and their spread among the spectrum, particularly in the green region, emphasized the need for refined evaluation metrics beyond R², suggesting a more complex approach to evaluating spectral matching. Our results show that the most popular measure does not provide an obvious result in selecting the best solution for the photometric application. The transition from simulations to the implementation phase confirmed this gap in the quality criteria. If a specific application is considered, the most popular spectral matching measure misses important information, which can be revealed only if the measurement data are used for modeling. If the shapes of the SPDs are carefully analyzed, high values of R² do not include information on the spectral distribution of the discrepancies, which is critical in some applications. That makes it difficult to select the best set of components without a very wide measurement procedure of a high number of sample components. That suggests the need for more refined evaluation metrics that can effectively capture the nuances of spectral fidelity beyond what R-squared values can provide and facilitate the design process. Such metrics should account for the locations of the component’s SPD mismatch and the photometric importance of different spectral regions.

This study serves as a step toward enhancing the reliability of the prediction of LED-based illuminants for photometric applications. Identifying gaps and searching for new modeling methods may contribute to improving the quality of predictions of solutions using LSTLS. Therefore, future research should focus on refining the modeling techniques, specifically adapting the quality criteria, to better predict the performance of the LSTLS and its components under varied operational conditions. Our main intention in the preliminary stage of the research was to underline the importance of the design criteria of the SPD of LSTLSs. As we showed in this manuscript, the most popular criteria do not allow for the best choice of SPD as many solutions have similar quality metrics values. The next step is the proposal of the methodology for the design of the LSTLS and the application of other criteria that are more suitable for the photometric design. Further steps will focus on the analysis of the impact of LED stability (degradation) and linearity, emphasizing the importance of LEDs with stable light flux responses to current and temperature changes, especially in precision applications. It is worth noticing that each LED emitter may degrade at a different rate due to various constructions and operating conditions, such as different supply currents and junction temperatures, even if the ambient temperature remains constant. These factors contribute to varying rates of optical power degradation over time. Therefore, the reliability of the stabilization systems (thermal and spectral feedback) will also be investigated. In our future research, we are committed to making significant contributions to the field of applied optoelectronics. The potential impact of this work includes refining simulation models, developing advanced evaluation metrics, and implementing methodological enhancements like machine learning techniques and real-time spectral feedback mechanisms. This forward-looking approach aims to address emerging challenges in LED photometry, and we believe it will significantly advance the field. Finally, the temperature management and stability of each channel’s control system are worth mentioning. Our goal is to comprehensively analyze reliability and stability (also in the life cycle) and provide guides for the holistic optimization of the LSTLS systems.