Optimising metameric spectra for integrative lighting to modulate the circadian system without affecting visual appearance

Approaches of evaluating the melanopic effectiveness of metameric spectra

The principle of univariance states that single cones are colour blind^42,43, meaning visual perception results from the antagonistic cone-opponency between the outer photoreceptors. This physiological phenomenon is the basis of metameric spectra, as different spectral power distributions can trigger the same visual response if the excitations across the photoreceptors do not change among the stimuli⁴³. In colourimetry, however, metameric spectra are defined as those whose chromaticity coordinates and luminance (or illuminance) match within a specified tolerance range. As the humans’ non-visual system works approximately independent of the visual pathway, it is possible to leverage such stimuli and modulate circadian responses without significantly affecting the chromaticity or luminous sensation. For example, at a CCT of 5552 K (Duv: 0.021, 11-channel), we optimised 386 spectra, whose chromaticity coordinates are located within the tolerance range of $\mathrm{\Delta }{u}^{\prime },\mathrm{\Delta }{v}^{\prime }\le 0.001$ (Fig. 1B, point 1). Subsequently, we used the melanopic EDI^34,36 $E{}_{\mathit{mel}}^{D65}$ to quantify the impact of the optimised metameric light spectra on the non-visual pathway, e.g. photic regulation of physiological and circadian responses in humans⁴⁴. This metric describes the melanopsin activated ipRGC response¹ without the extrinsic synaptic input from outer retinal photoreceptors to the ipRGCs^14,45. According to common interior lighting recommendations²¹, we considered those spectra that fulfil a specified colour rendering metric, which is in Fig. 1B (point 2), for example, a CRI R_a higher than 80. From the set of optimised spectra, the upper and lower limit of the melanopic EDI with $E{}_{mel,Max}^{D65}$ and $E{}_{mel,Min}^{D65}$ can be extracted (Fig. 1B, point 2). From the remaining two metameric pairs, we calculated the maximum possible change of the melanopic EDI for this specific chromaticity point with $\mathrm{\Delta }{E}_{\mathit{mel}}^{D65}=|E_{mel,Max}^{D65}-E_{mel,Min}^{D65}|$. As shown in Fig. 1B point 2, using a CCT of 5552 K and Duv of 0.021, the melanopic EDI can be varied from 181 to 271 lx ($\mathrm{\Delta }E{}_{\mathit{mel}}^{D65}\approx$ 90 lx), corresponding to a melanopic Michelson (see “Methods”) contrast $C_M$ of ~ 0.20, although the photopic illuminance remains steady at 250 lx. Previous works on optimising highly effective non-visual spectra^30,39,46 aimed to catch the $E{}_{mel,Max}^{D65}$ stimuli or other metrics that correlate with the upper limit, while in metameric optimisation, the lower limit must also be evaluated to assess the maximum tuning range $\mathrm{\Delta }E{}_{\mathit{mel}}^{D65}$ and its corresponding melanopic Michelson contrast $C{}_M$ for a particular chromaticity and photopic illuminance.

Apart from stating the tuning range of the melanopic illuminance, the ratio between $E{}_{\mathit{mel}}^{D65}$ and $E_v$ is a standard metric for assessing the melanopic efficacy of spectra, denoted as the melanopic daylight efficacy ratio $\gamma {}_{\mathit{mel}}^{D65}$ (melanopic DER)^14,36. The value $\gamma {}_{\mathit{mel}}^{D65}$ can be used as a factor to calculate $E{}_{\mathit{mel}}^{D65}$ as a function of any photopic illuminance. If, for example, $\gamma {}_{\mathit{mel}}^{D65}$ is 0.8 for a given spectrum, then a photopic illuminance of 500 lx could achieve a $E{}_{\mathit{mel}}^{D65}$ of 400 lx, calculated with the formula $E{}_{\mathit{mel}}^{D65}=E_v·\gamma {}_{\mathit{mel}}^{D65}$. By knowing a melanopic DER value, the photopic illuminance that is required to get a specific $E{}_{\mathit{mel}}^{D65}$ can be calculated. This makes $\gamma {}_{\mathit{mel}}^{D65}$ a highly useful quantity for scientific analyses, as the metric provides a generalised description uncoupled from a pre-specified photopic illuminance level. Therefore, instead of expressing the melanopic tuning of a particular chromaticity point in terms of $\mathrm{\Delta }E{}_{\mathit{mel}}^{D65}$, it is a good scientific practise to express the tuning range in terms of its $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ value, which can be calculated using the upper and lower limit of the melanopic DER according to $|{\gamma }_{mel,Max}^{D65}-{\gamma }_{mel,Min}^{D65}|$ (see Fig. 1B, point 2). The tuning range in melanopic illuminance $\mathrm{\Delta }E{}_{\mathit{mel}}^{D65}$ for a particular photopic illuminance $E{}_v$ can be calculated using the formula $\mathrm{\Delta }E{}_{\mathit{mel}}^{D65}=E_v·\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$. Note that a melanopic DER $\gamma {}_{\mathit{mel}}^{D65}$ greater than one indicates that the melanopic equivalent daylight (D65) illuminance is higher than the (photopic) illuminance of the source. For the standard illuminant D65, the melanopic DER equals to one. It should be mentioned, that knowing $\mathrm{\Delta }E{}_{\mathit{mel}}^{D65}$ of metameric pairs is not sufficient to predict circadian responses. For instance, melatonin suppression is a function of $E{}_{\mathit{mel}}^{D65}$ and follows a sigmoidal function that saturates at high melanopic equivalent daylight illuminances³⁴. Thus, in addition to the melanopic tuning with $\mathrm{\Delta }{E}_{\mathit{mel}}^{D65}=|E_{mel,Max}^{D65}-E_{mel,Min}^{D65}|$, the starting point $E{}_{mel,Min}^{D65}$ is of interest for evaluating the circadian effectiveness, which indicates to what extend the melatonin suppression can be modulated without altering visual metrics.

In this work, however, the spectral dataset of metamer spectra was optimised for each chromaticity target from which a spectrum with $E{}_{mel,Min}^{D65}$ and one with $E{}_{mel,Max}^{D65}$ can be identified. Therefore, the melanopic DER can be calculated for the upper and lower limit of $E{}_{\mathit{mel}}^{D65}$ with $\gamma {}_{mel,Max}^{D65}$ and $\gamma {}_{mel,Min}^{D65}$ for each chromaticity target (Fig. 1B, point 2). Additionally, the melanopic EDI of the two metameric pairs can be used to calculate the melanopic Michelson contrast $C_M$ (see “Methods”). As numerous spectra were calculated for each chromaticity target, we can even evaluate the spectral optimisation limit for non-visual purposes, making it possible to state for which Duv value of a CCT the highest $\mathrm{\Delta }E{}_{\mathit{mel}}^{D65}$ (or generalised $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65})$ can be achieved (Fig. 1C).

For each CCT, numerous chromaticity coordinate targets were used, meaning different Duv distances to the Planckian locus (Fig. 1C). One essential question in optimisation procedures is, which chromaticity coordinate should be chosen along the Duv distances to reach a spectrum with the largest melanopic metameric tuning range. As we determined the $E{}_{mel,Min}^{D65}$ and $E{}_{mel,Max}^{D65}$ values for different chromaticity points, the upper and lower limits of the melanopic DER ($\gamma {}_{mel,Max}^{D65}$, $\gamma {}_{mel,Min}^{D65}$) is known across the Duv steps for each CCT. In Fig. 1C, it can be observed that with a Duv of − 0.018 (2901 K), the largest melanopic DER change can be achieved, resulting in a $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ of 0.29. For example, when using a photopic illuminance of 250 lx, this would correspond to a metameric melanopic EDI change $\mathrm{\Delta }E{}_{\mathit{mel}}^{D65}$ of ~ 73 lx (see Fig. 1C, middle panel). However, if the objective is to optimise spectra with the largest non-metameric melanopic efficacy $\widehat{\gamma }{}_{mel,Max}^{D65}$ at a particular fixed CCT, a chromaticity coordinate with a Duv of -0.039 should be preferred as an optimisation target, leading to a $\gamma {}_{mel,Max}^{D65}$ of 0.74, but whose metameric tuning $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ value is lower with 0.236 ($\mathrm{\Delta }E{}_{\mathit{mel}}^{D65}\approx 59$ lx). Thus, when selecting the ideal chromaticity coordinate (fixed CCT) for spectral optimisation, it must be decided whether the objective is to maximise or minimise the melanopic DER $\gamma {}_{\mathit{mel}}^{D65}$ or to find a metameric pair with the largest melanopic tuning range $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$. In the following, the largest melanopic DER of a particular CCT will be denoted as $\widehat{\gamma }{}_{mel,Max}^{D65}$, its lowest melanopic DER as $\widehat{\gamma }{}_{mel,Min}^{D65}$ and its largest melanopic tuning range across all Duv values as $\mathrm{\Delta }\tilde{\gamma }{}_{\mathit{mel}}^{D65}$ (see Fig. 1C and D).

As discussed, the objective of this work is to analyse the melanopic tuning range of metameric spectra while ensuring a good colour rendition. The colour rendition of a light source is an essential aspect of integrative lighting and can influence the subjectively evaluated naturalness, vividness and preference of an object’s colour in an illuminated environment⁴⁷. In this work, we evaluated the colour rendition of the optimised light spectra by considering only the aspect of colour fidelity, using the ANSI/IES colour fidelity index R_f (TM-30-20) as metric⁴⁰. The colour fidelity index R_f is in its characterisation intent analogous to the general colour rendering index⁴⁷ (CRI) R_a. However, the latter (CRI R_a) has several shortcomings when evaluating artificial polychromatic light that is mixed with narrowband LED spectra (see the works^47–52 for further explanation), which is why the colour fidelity index R_f is recommended in the CIE 224:2017⁵³ for scientific use cases. To select appropriate R_f threshold conditions, we have followed the TM-30-20 Annex E recommendations⁴⁰, which specify distinct criteria for the categories colour preference, colour vividness, and colour fidelity, separated into three priority levels, respectively^40,47. These criteria can be applied for polychromatic LED spectra in illuminated environments with a photopic illuminance between 200 and 700 lx^40,47. The intended design goal in this work is to ensure a good colour fidelity of the optimised light spectra. For this, we have chosen the priority level three (R_f $\ge$ 85, R_f,h1 $\ge$ 85) and level two (R_f $\ge$ 90, R_f,h1 $\ge$ 90) from the fidelity category, where R_f,h1 is defined as the local colour fidelity of the first bin in the colour vector graphic⁴⁰. In the next section, the analysis of the optimised spectra will be carried out with the discussed strategy and metrics. To state the melanopic DER range for a particular CCT, we plotted all computed melanopic DER values for the various Duv steps of a single CCT as a single category (Fig. 1D).

The melanopic DER tuning range of metameric spectra increases with higher CCTs

For each of the investigated chromaticity targets (see Fig. 1A), we calculated the maximum melanopic tuning range, which we defined as $\mathrm{\Delta }{\gamma }_{\mathit{mel}}^{D65}=|{\gamma }_{mel,Max}^{D65}-{\gamma }_{mel,Min}^{D65}|$. Accordingly, the melanopic tuning range $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ was mapped in the CIExy colour space for each defined fidelity criterium (Fig. 2A). The results indicate that, regardless of the LED channel number, chromaticity coordinate and fidelity criterium, the largest melanopic tuning range $\mathrm{\Delta }\gamma {}_{mel,Max}^{D65}$ across all CCTs varies between 0.14 to 0.30 (Fig. 2A). This tuning range can be achieved for the different luminaire configurations without altering visual metrics such as chromaticity or photopic illuminance. Furthermore, the largest melanopic tuning range across all CCTs can be reached for chromaticity coordinates that are above Planck (Duv 0 to 0.006) and correspond to the CCTs 6702 K (6-channel, Duv: 0.003), 7443 K (8-channel, Duv: 0) and 6702 K (11-channel, Duv: 0.006) if a colour fidelity condition of R_f $\ge$ 85 and R_f,h1 $\ge$ 85 (priority level three) is applied (see red dots in Fig. 2A).

However, with rising CCT, the melanopic tuning range $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ increases, while a saturation effect is notable from approximately ~ 4500 K onwards, meaning that at a specific CCT threshold that additionally depend on the LED channel number, $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ only marginally increases (see Supplementary materials, Fig. S2). Note that the melanopic tuning range at steady CCT differs between the various Duv values (see Fig. 2A). When using an 8-channel or 11-channel LED luminaire (R_f $\ge$ 85, R_f,h1 $\ge$ 85), the (Duv) area in which a melanopic tuning range $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ of at least 0.2 ($\mathrm{\Delta }E{}_{\mathit{mel}}^{D65}$ = 50 lx for $E_V$= 250 lx) can be reached increases (Fig. 2A). If the colour fidelity criterion is set to the priority level two (R_f $\ge$ 90, R_f,h1 $\ge$ 90), then the largest melanopic tuning range $\mathrm{\Delta }\gamma {}_{mel,Max}^{D65}$ across all CCT steps remains between 0.14 and 0.19, depending on the used LED configurations (Fig. 2A). However, from about 3000 K upwards, the $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ saturates and becomes approximately independent of the CCT when using the second priority level (R_f $\ge$ 90, R_f,h1 $\ge$ 90) of the fidelity criterion (Supplementary materials, Fig. S2). In Fig. 2B, the relative metamer spectra pairs that could cause the largest melanopic tuning range $\mathrm{\Delta }\gamma {}_{mel,Max}^{D65}$ are shown (see red dots in Fig. 2A). It merits to be noted that the melanopic DER (and hence the melanopic EDI) can be tuned by increasing the blue LED-channel’s duty cycle (${\lambda }_{\mathit{Peak}}=470$ nm) and adapting the other (opposite) channels to maintain the chromaticity point and photopic illuminance (Fig. 2B).

The $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ value is an intuitive way of visualising the melanopic tuning range for various metameric spectra. However, for evaluating the impact of a melanopic modulation and not only the $\gamma {}_{\mathit{mel}}^{D65}$ value at a particular CCT, it is important to also check the relative contrast of the melanopic EDI between the highest and lowest melanopic condition of a metameric pair. For this, we additionally mapped the melanopic Michelson contrast $C_M$ (see “Methods”) in the CIExy-2° colour space to visualise those chromaticity coordinate regions where spectra can be optimised that feature the highest relative change related to $E{}_{mel,Min}^{D65}$ (Fig. 3A, B).

The largest melanopic contrast can be considered either isolated for each CCT or globally across all CCT and Duv steps. The latter results in a $C_{M,Max}$ spot (see “Methods”), similar to the maximum in the previous analysis in (red point in Fig. 2A) for $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$. The $C_{M,Max}$ peak contrast value reflects the melanopic modulation limit across all optimised chromaticity targets when using metameric spectra, indicating the technical possibility of a multi-channel LED luminaire configuration. Depending on the number of channels, the maximum melanopic Michelson contrast $C_{M,Max}$ across all CCT and Duv steps that can be reached, is between 0.16 and 0.18 when considering only those spectra that fulfil the third priority level (R_f $\ge$ 85, R_f,h1 $\ge$ 85) of the colour fidelity (Fig. 3A). For example, with the 6-channel LED configuration (Fig. 3A), a $C_{M,Max}$ of 0.16 can be realised below the Planckian locus with a CCT of 3017 K (Duv − 0.018), allowing to modulate the melanopic EDI from approximately 135 lx to 185 lx ($\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}\approx 0.2$) without affecting the photopic illuminance (250 lx) and the chromaticity (Fig. 3B). With the 8-channel luminaire, a maximum melanopic Michelson contrast of 0.18 is possible at a CCT of 3456 K (Duv 0.009), with which the melanopic EDI can be modulated from approximately 115 lx to 165 lx, corresponding to a $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ of $\sim$ 0.2 (Fig. 3B). When using an 11-channel luminaire, the $C_{M,Max}$ peak is achieved at a CCT of 3456 K (Duv 0.009), resulting in the ability to modulate the melanopic EDI from 115 to 165 lx, corresponding to a $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$ of ~ 0.2 (Fig. 3B). Thus, the level of the melanopic contrast $C_{M,Max}$ (red point in Fig. 3A) is not affected between the 8-channel and 11-channel luminaire at the priority level three of the colour fidelity (R_f $\ge$ 85, R_f,h1 $\ge$ 85). Note that the reported maximum possible contrast across all CCTs or for an isolated CCT is given under our methodological condition, meaning that the metameric spectra must be within a distance of $\mathrm{\Delta }{u}^{\prime },\mathrm{\Delta }{v}^{\prime }\le 0.001$ from the chromaticity target. Slightly higher contrasts can be achieved if a larger tolerance to the chromaticity coordinate target is chosen, as this results in a larger number of spectra that can be considered for generating the metameric Michelson contrast. The applied tolerance of $\mathrm{\Delta }{u}^{\prime },\mathrm{\Delta }{v}^{\prime }\le 0.001$ was chosen so that the ellipses (Fig. 1B, step 2) around each chromaticity coordinate did not overlap, as we used a tight grid of optimisation targets along the Planckian locus (Fig. 1A).

The projection of the metamerism contrast values $C_M$ into the CIExy colour space (Fig. 3A) shows distinct areas in which a certain level of melanopic modulation can be achieved. For example, using the 6-channel luminaire, metameric spectra (R_f $\ge$ 85, R_f,h1 $\ge$ 85) with a $C_M$ of at least 0.14 can be generated when choosing CCTs between 2700 to 4717 K within a Duv distance to Planck between − 0.027 to 0.015. However, by increasing the luminaire’s number of channels to eight, the chromaticity colour area in which a melanopic contrast of $C_M$ > 0.14 (R_f $\ge$ 85, R_f,h1 $\ge$ 85) can be achieved widens (Fig. 3A). With the 11-channel LED luminaire, isolated islands of chromaticity regions between 2700 and 5102 K (Duv − 0.018 to 0.027) can be identified with which a melanopic Michelson contrast $C_M$ between 0.16 to 0.18 (R_f $\ge$ 85, R_f,h1 $\ge$ 85) can be optimised. In the supplementary materials (Figs. S3, S4), we provide the maps of the non-visual metameric modulation limits without any colour fidelity condition, showing that the colour rendering is the major limiting factor when designing metameric spectra for non-visual purposes.

Comparing the melanopic efficacy of metameric and non-metameric spectra

In this section, we analyse the determined melanopic DER $\gamma {}_{\mathit{mel}}^{D65}$ values for both, $E{}_{mel,Min}^{D65}$ and $E{}_{mel,Max}^{D65}$ as a function of CCT (Fig. 1D). For each CCT, multiple $\gamma {}_{mel,Min}^{D65}$ (black scatter) and $\gamma {}_{mel,Max}^{D65}$ (red scatter) values exist (Fig. 4A) since spectra were optimised for each CCT with about 32 different Duv distances (i.e., chromaticity targets per CCT, see Fig. 1 for explanation). With higher CCT, the distance between $\gamma {}_{mel,Min}^{D65}$ and $\gamma {}_{mel,Max}^{D65}$ distributions increases, and the respective scatter becomes narrower (Fig. 4A). Therefore, we can state, that above ~ 7000 K, the selection of the chromaticity coordinate (at steady CCT) does not significantly affect the melanopic efficacy of a spectral stimulus. At lower CCTs (< ~ 4500 K), highly circadian effective spectra could only be achieved with selected chromaticity coordinates.

This analysis (Fig. 4A) applies for non-metameric spectra, in which the Duv is a free parameter for each CCT. Such an aspect is crucial for comparing the melanopic tuning range of metameric spectra (steady chromaticity, Fig. 3) to non-metameric spectra with a fixed CCT, but variable Duv distance to Planck. From approximately 4600 K onwards (Fig. 4A), a spectrum can be generated that reaches a melanopic DER greater than one, regardless of the luminaire’s LED setting (R_f $\ge$ 85, R_f,h1 $\ge$ 85). However, when applying a stricter colour fidelity index condition (R_f $\ge$ 90, R_f,h1 $\ge$ 90), the CCT threshold with which a melanopic DER greater than one can be achieved is shifted towards higher CCTs to approximately ~ 6000 K (Fig. 4A). In lighting applications, it is often of interest to vary the melanopic efficacy at given CCT with no restriction on the choice of the Duv distance to the Planckian locus⁵⁴, which would be not metameric but could result in the advantage of reaching higher melanopic efficacy. Therefore, we also plotted the trend line in Fig. 4A to denote the lowest melanopic DER value, $\widehat{\gamma }{}_{mel,Min}^{D65}$ and the largest value $\widehat{\gamma }{}_{mel,Max}^{D65}$ that can be generated for each CCT for non-metameric spectra. Note that the span width between the $\widehat{\gamma }{}_{mel,Min}^{D65}$ and $\widehat{\gamma }{}_{mel,Max}^{D65}$ lines in Fig. 4A should not be interpreted as metameric tuning range parameter with $\mathrm{\Delta }\gamma {}_{\mathit{mel}}^{D65}$. For the latter only the highest and lowest melanopic DER for a particular Duv and CCT value needs to be considered (i.e. truly metameric spectra, see Fig. 1C, D).

From a practical point of view, the question arises which chromaticity coordinates between 2700 to 7443 K should be used to obtain spectra at the upper limit of $\gamma {}_{mel,Max}^{D65}$ (red line in Fig. 4A), as the melanopic efficacy of a spectrum could depend on its Duv distance to Planck. In Fig. 4B, the optimal chromaticity coordinates for spectral optimisation are highlighted (red points), making it possible to obtain the $\widehat{\gamma }{}_{mel,Max}^{D65}$ values (non-metameric). The ideal chromaticity coordinates for reaching $\widehat{\gamma }{}_{mel,Max}^{D65}$ are below the Planckian locus (6-channel: Duv range: -0.039 to -0.009). When applying the priority level two criterion for the colour fidelity index (R_f $\ge$ 90, R_f,h1 $\ge$ 90), the ideal chromaticity targets for $\widehat{\gamma }{}_{mel,Max}^{D65}$ (6-channel) move closer to the Planckian locus within a Duv area between − 0.036 to − 0.006. As discussed in the first section, it must be assumed that the chromaticity regions that provide spectra with a maximum melanopic efficacy $\widehat{\gamma }{}_{mel,Max}^{D65}$ could differ from those that could achieve the largest melanopic tuning range $\mathrm{\Delta }\tilde{\gamma }{}_{mel,Max}^{D65}$ through metameric spectra (Fig. 1C). For example, in Fig. 4B, the chromaticity coordinates (blue points) are highlighted with which metameric spectra can be optimised to reach $\mathrm{\Delta }\tilde{\gamma }{}_{mel,Max}^{D65}$ for each CCT. These chromaticity points are closer to the Planckian locus within a Duv area between -0.027 and 0.009 (6-channel, R_f $\ge$ 85, R_f,h1 $\ge$ 85). In contrast, the chromaticity location with which the lowest melanopic DER $\widehat{\gamma }{}_{mel,Min}^{D65}$ can be achieved (green points) are above Planck, regardless of the applied colour fidelity condition. Based on these results, however, we can state that the melanopic efficacy of the spectra depends significantly on the selected chromaticity coordinates and that spectra with a highly melanopic efficacy (non-metameric) are more likely to be found below the Planckian locus.

Multi-channel LED luminaire

The primary spectra of an existing 11-channel LED luminaire (Thouslite LEDCube) were measured using a Konica Minolta CS2000 spectroradiometer as a function of the duty-cycle and then interpolated. From the interpolated channel-spectra, a 6-channel, 8-channel and 11-channel LED luminaire was simulated in MATLAB MathWorks. The 6-channel LED luminaire had four narrowband LEDS with peak-wavelengths of 475 nm, 504 nm, 521 nm, 662 nm and two phosphor-converted white LEDs with a CCT of 4655 K and 2740 K. The 8-channel LED luminaire had narrowband LEDs with peak wavelengths of 450 nm, 465 nm, 504 nm, 521 nm, 638 nm, 662 nm and two phosphor-converted white LEDs (4655 K, 2740 K). The 11-channel LED luminaire had 8 narrowband LEDs with peak wavelengths of 419 nm, 450 nm, 457 nm, 504 nm, 521 nm, 597 nm, 638 nm, 662 nm, one phosphor-converted lime coloured LED and two white LEDs (4655 K, 2740 K).

The primary spectra of each luminaire configuration are reported in the appendix (Supplementary materials, Fig. S1). The selection of the LED combinations was made in such a way that with an increasing number of channels, more chromatic LEDs were added, which should result in a higher degree of freedom to maintain a chromaticity coordinate while modulating other melanopic metrics. During optimisation, the number of channels was specified as an adjustable parameter. This means that not necessarily all primary spectra are to be used. However, within a solution, the maximum number of used channels does not exceed the set limit. For example, it is possible that when using the 11-channel configuration, there could also be spectra/solutions that activate 9-channels rather than 11-channel.

Spectral optimisation and metameric calculations

A uniform grid of 561 chromaticity coordinates (2700 K to 7443 K ± Duv 0 to 0.048) along the Planckian locus in the CIEu’v’-1976 colour space was used to define optimisation objectives (see Supplementary materials Fig. S1), similar to the authors’ previous publication^19,41. For each chromaticity coordinate, spectra were optimised in a loop with six to eight repetitions, making sure to calculate enough metamers.

The spectra were optimised with the tolerance condition $\mathrm{\Delta }{u}^{\prime },\mathrm{\Delta }{v}^{\prime }\le 0.001$ for the chromaticity targets. Thus, spectra within $\mathrm{\Delta }{u}^{\prime }{v}^{\prime }$ $\le$ 1.41 $·{10}^{-3}$ were defined as metamers, leading to a total of 490,068 optimised spectra. The spectra $X(\lambda )$ were initially optimised for an illuminance of 220 lx ± 2 lx. Then, all optimised spectra were linearly (re)scaled to reach a fixed photopic illuminance $E_v$ of 250 lx.

All optimisation results and scripts are available at the corresponding author’s GitHub repository. The spectra were optimised using a custom-developed heuristic optimisation procedure, which will be published separately. However, metameric spectra can be generated with any heuristic optimisation or by using other approaches as extensively discussed in detail by numerous publications and reviews^{30–32,64,69,77–80}.

From the optimisation, a set of $N$ metamer spectra $M_S=[X_1(\lambda ),X_2(\lambda ),\cdots ,X_N(\lambda )]$ for $X_1(\lambda ),X_1(\lambda ),\cdots ,X_N(\lambda )\in {\mathbb{R}}^{401\times N\times C}$ with $C$ as the number of target chromaticity coordinates was derived. The non-visual impact of the optimised spectra $M_S$ was assessed by quantifying the melanopic stimulus strength in terms of melanopic EDI ${\{E_{mel,i}^{D65}\}}_{i=1}^C$ for $E_{mel,i}^{D65}ϵ{\mathbb{R}}^{N\times 1}$, using the formula (Eq. 1)

with $s_{\mathit{mel}}(\lambda )$ as the melanopic action spectrum³⁶ according to the CIE 026/E:2018. Next, for each chromaticity coordinate (i), the maximum melanopic tuning range was calculated across the metameric spectra with $\mathrm{\Delta }{E}_{mel,i}^{D65}=|\max E_{mel,i}^{D65}-\min E_{mel,i}^{D65}|$ (see Fig. 1). As the melanopic EDI tuning range depends on the applied photopic illuminance, we additionally calculated the melanopic DER according to (Eq. 2)

where $E_v$ is defined as our used photopic illuminance of 250 lx. The $\mathrm{\Delta }\gamma {}_{mel,i}^{D65}$ values were interpolated and plotted using a heatmap in the CIExy-2° colour space (Fig. 2A). The spot with the highest possible melanopic tuning range across all chromaticity coordinates was calculated with $\mathrm{\Delta }\gamma {}_{mel,Max}^{D65}=\max \mathrm{\Delta }\gamma {}_{mel,i}^{D65}$. The respective $\mathrm{\Delta }\gamma {}_{mel,Max}^{D65}$ values were marked as red dot in Fig. 2A. The metrics of the $\mathrm{\Delta }\gamma {}_{mel,Max}^{D65}$ spectra are additionally stated in the supplementary materials for each luminaire configuration and colour fidelity condition. In Fig. 3A, we also plotted the melanopic Michelson contrast C_M,i, which was calculated using the formula (Eq. 3)

where $E{}_{mel,Max,i}^{D65}$ corresponds to $\max E{}_{mel,i}^{D65}$ and $E{}_{mel,Min,i}^{D65}$ to $\min E{}_{mel,i}^{D65}$ from the optimised metameric set $M_S$ of each chromaticity target. The procedure of calculating derived metrics is discussed in the first section of this manuscript (see Fig. 1). For calculating the IES TM-30-20 colour fidelity metrics from the spectra, we used the LuxPy library⁸¹, developed by Kevin Smet. All optimized spectra, the calculated metrics, the MathWorks MATLAB and Python scripts are available online at the main author’s GitHub repository (see data availability statement).