Visual Performance and Photobiological Effects of White LED Systems Based on Spectral Compensation

Abstract

The visual performance and photobiological effects of white LED systems based on spectral compensation are discussed, specifically focusing on the total optical power, the ratio of scotopic vision luminous flux to photopic vision luminous flux (S/P), the blue light hazard (BLH), and the circadian action factor (CAF). Theoretical models are established by integrating the spectral power distribution (SPD) with spectral sensitivity functions associated with the human visual system, and meanwhile, the impacts of LEDs’ electro-thermal characteristics on the mixed spectral structure and optical properties are analyzed. As experimental results demonstrate, an excellent agreement is shown between the calculated and measured values of the total optical power, S/P, BLH, and CAF, in terms of both values and variation trends. These proposed models are expected to serve as effective tools for understanding the visual perception and non-visual biological effects in specific illumination environments. Moreover, they can offer valuable reference frameworks for the development of lighting solutions that are more human-centered and health-oriented.

1. Introduction

With the rapid development of solid-state lighting technology, light-emitting diodes (LEDs) have emerged as the fourth-generation lighting sources, attributed to their high luminous efficiency, color-tunable property, and environmental friendliness [1,2]. Over the past few decades, the design, optimization, and control of LED devices and systems have predominantly centered around visual performance, aiming to enhance luminous efficiency and color quality [3,4]. However, with the increasing duration of people’s exposure to artificial lighting environments, findings from biological and medical research indicate that light stimuli have significant photobiological impacts on humans, which are also commonly referred to as non-visual biological effects [5,6,7]. The potential health risks posed by blue light exposure have garnered growing research interest since 1966 [8]. Characterized by relatively high energy, blue light can penetrate the cornea and lens and reach the retina directly, causing photochemical damage to the retina, which is called blue light hazard (BLH) [9]. It has been proved that photochemical damage due to excessive ocular exposure to blue light can trigger a cascade of pathological changes, including oxidative stress, mitochondrial dysfunction-mediated apoptosis, inflammatory responses, and DNA damage [10,11,12]. The human retina contains various photoreceptor cells to receive light stimuli. Among them, the intrinsic photosensitive retinal ganglion cells (ipRGCs), are mainly involved in regulating physiological functions such as work efficiency, circadian rhythm, reaction time and cognitive performance [13,14]. Even more, inappropriate light stimuli may disrupt physiological functions, leading to multiple negative impacts on human health, though not all of these impacts have been fully elucidated [15,16]. The circadian action factor (CAF) proposed by Gall et al. is used to quantify the role of lighting in regulating human physiological functions, and now it has been gradually adopted as a quantitative metric in evaluation and design of light sources [17].

As the health consciousness continues to grow, there is a rising preference for natural lighting, especially in indoor environments [18]. Consequently, it is essential to take into account both the visual and non-visual biological effects of light sources on humans to create human-centric and healthy lighting environments [19]. The phosphor-converted (PC) white LEDs, which combine YAG:Ce³⁺ phosphors and blue GaN-based chips, are the dominant artificial lighting sources due to their technological maturity and cost-effectiveness [20,21]. Nonetheless, these PC white LEDs have a relatively fixed spectral structure, lacking red and green spectra [22,23]. As a result, they typically exhibit a nearly constant correlated color temperature (CCT) and a relatively low color rendering index (CRI) [24]. In our prior research, significant advancements were achieved in investigating the potential of PC white LEDs optimized through red and green spectral compensation. The study validated that this compensation methodology could effectively modulate the CCT while concurrently improving the CRI, thereby enhancing the overall lighting quality [25]. Notwithstanding these findings, the published work did not fully explore the visual and photobiological impacts of these compensated white LED systems, primarily due to spatial limitations.

In view of the above-mentioned issues, this paper focuses on the compensated white LED systems composed of a PC white LED, a monochromatic green LED, and a monochromatic red LED, all of which can be independently driven. According to the tunable dimming solution proposed linking the spectral colorimetry and the general photo-electro-thermal (PET) theory, the core objective of this paper is to explore the visual performance and photobiological effects of the LED systems based on spectral compensation. Through detailed discussions of theoretical analysis and experimental verification, it aims to comprehensively understand the practical significance of spectral compensation in LED lighting applications, to clarify the relationships between the LEDs’ electro-thermal characteristics and their impacts on visual and photobiological effects, and ultimately contribute to the development of lighting solutions that are more human-centered and health-oriented.

2. Analysis and Modeling

Human visual perception is a complex process, which is closely related to the spectral structure and characteristics of light sources [26]. Under varying lighting conditions, the human visual system exhibits different capabilities in discerning details and perceiving colors [27]. Correspondingly, the physiological functions regulated by light stimulus also differ under diverse lighting scenarios. In our prior research [28], two crucial parameters CCT and CRI of the compensated white LED systems have been studied. To further assess lighting from a human-centered perspective, this study conducts an in-depth exploration of several additional aspects: the total optical power, the ratio of scotopic vision luminous flux to photopic vision luminous flux (S/P), the blue light hazard (BLH), and the circadian action factor (CAF).

It is noted that the compensated white LED system studied in this paper is composed of a PC white LED, a monochromatic red LED, and a monochromatic green LED—all of which support independent driving. A structural schematic of this system is provided in Figure 1. In this system, the PC white LED serves as the core component and operates in a steady state, thereby providing the basic luminous flux required for lighting applications. Meanwhile, the monochromatic red and green LEDs function as spectral compensators, enabling adjustment of red and green spectral components by regulating the electrical driving state.

2.1. The Total Optical Power (Popt)

As the compensated white LED system is composed of three LEDs, i.e., a PC white LED, a monochromatic red LED, and a monochromatic green LED, the total SPD and optical power thereby can be described as

$$\left\{\begin{array}{c}P\left(\lambda \right)=P_W\left(\lambda \right)+P_R\left(\lambda \right)+P_G\left(\lambda \right)\\ P_{opt}=P_{opt,W}+P_{opt,R}+P_{opt,G}\end{array}\right.$$

(1)

where P_W(λ), P_R(λ) and P_G(λ) are SPDs corresponding to the PC white LED, the monochromatic red LED, and the monochromatic green LED, respectively, and P(λ) is the total SPD of the mixed spectra; P_opt,W, P_opt,R and P_opt,G are optical power corresponding to the PC white LED, the monochromatic red LED, and the monochromatic green LED, respectively, and P_opt is the total optical power of the mixed spectra.

For a monochromatic LED under an effective thermal design, a linear relationship exists between the optical power and the driving current [29], as clearly shown in Figure 2a. Consequently, for both the monochromatic red LED and the monochromatic green LED of the compensated white LED system, their optical power can be mathematically expressed as a linear function of the current,

$$\left\{\begin{array}{c}{P}_{opt,R}\left(I_R\right)=k_{R1}+k_{R2}{I}_R\\ P_{opt,G}\left(I_G\right)=k_{G1}+k_{G2}{I}_G\end{array}\right.$$

(2)

where k_R1 and k_R2 are the coefficients describing the relationship between the optical power and driving current, for the monochromatic red LED; correspondingly, k_G1 and k_G2 are the coefficients describing the relationship between the optical power and driving current, for the monochromatic green LED.

Since the PC white LED is operating in a steady state, and the monochromatic red and green LEDs are driven with changing currents, by combining Equations (1) and (2), the total optical power of the compensated white LED system is further described as

$$\begin{gathered} P_{opt}\left(I_R,I_G\right)=P_{opt,W0}+\left(k_{R1}+k_{R2}{I}_R\right)+\left(k_{G1}+k_{G2}{I}_G\right) \\ =P_{opt,W0}+\left(k_{R1}+k_{R2}{I}_R\right)+\left(k_{G1}+k_{G2}{I}_G\right) \\ ={\displaystyle \frac{\left(k_{WRG0}+k_{R2}{I}_R\right)\left(k_{WR0G}+k_{G2}{I}_G\right)}{P_{opt,0}}} \end{gathered}$$

(3)

where k_R1, k_R2, k_G1 and k_G2 are the same as defined in Equation (2); P_opt,W0 represents the constant optical power of the PC white LED; P_opt,0 means the total optical power of the compensated white LED system at the given operating point (I_R,0, I_G,0), i.e., the red LED is operating with current I_R,0 and the green LED is operating with current I_G,0, and specifically P_opt,0 = P_opt,W0 + k_R1 + k_R2 I_R0 + k_G1 + k_G2 I_G0; k_WR0G and k_WRG0 are merged parameters introduced in the model to simplify the expression, respectively, of k_WR0G = P_opt,W0 + k_R1 + k_R2I_R0 + k_G1, k_WRG0 = P_opt,W0 + k_R1 + k_G1 + k_G2I_G0.

Equation (3) indicates that the total optical power of the mixed spectra emitted from the compensated white LED system can be expressed as a function of the driving currents I_R and I_G, where I_R and I_G correspond to the driving currents of the monochromatic red LED and monochromatic green LED, respectively.

2.2. The Ratio of Scotopic Vision Luminous Flux to the Photopic Vision Luminous Flux (S/P)

The retinal cells in human eyes exhibit distinct differences in light sensitivity and color perception, enabling humans to adapt to a wide range of ambient lighting conditions and efficiently gather visual information [30]. Photopic vision occurs in bright environments, where human eyes predominantly rely on cone cells to perceive light stimulus and distinguish object details such as color, shape, and texture. In contrast, scotopic vision is associated with darker environments, where rod cells become active in response to light stimuli. It is noted that although rod cells are highly sensitive to light, they have extremely limited color-discrimination ability. Actually, the major contribution of these rod cells is to assist in identifying the outlines and general shapes of objects in the dark, facilitating the perception of the surrounding environment.

The S/P represents the ratio of the scotopic vision luminous flux (S) to the photopic vision luminous flux (P), which is utilized to assess the luminous efficacy of light sources at low luminance levels [31]. In lighting design, the S/P can, on one hand, reflect the adaptability of the human visual system to the variation in illumination; on the other hand, it can serve as a valuable reference for adjusting the parameters of light sources and the surrounding environment according to specific visual requirements [32]. S/P is defined as

$$S/P={\displaystyle \frac{K_s{\int }_{380}^{780}P\left(\lambda \right)V^{\prime }\left(\lambda \right)d\lambda }{K_m{\int }_{380}^{780}P\left(\lambda \right)V\left(\lambda \right)d\lambda }}$$

(4)

where P(λ) is the spectral power distribution (SPD) of the light source; V′(λ) and V(λ) are the spectral sensitivity functions under scotopic vision and photopic vision, respectively, with peak wavelengths corresponding to 507 nm and 555 nm; K_s and K_m are defined as the maximum spectral luminous efficacy values for scotopic vision (1700 lm/W) and photopic vision (683 lm/W), respectively [32].

Referring to Equation (1), the SPD of the compensated white LED system is given by P(λ) = P_W(λ) + P_R(λ) + P_G(λ); thus, Equation (4) can be rewritten as

$$S/P={\displaystyle \frac{K_s{\int }_{380}^{780}\left[P_W\left(\lambda \right)+P_R\left(\lambda \right)+P_G\left(\lambda \right)\right]V^{\prime }\left(\lambda \right)d\lambda }{K_m{\int }_{380}^{780}\left[P_W\left(\lambda \right)+P_R\left(\lambda \right)+P_G\left(\lambda \right)\right]V\left(\lambda \right)d\lambda }}$$

(5)

Typically, the SPD of a monochromatic LED can be described using a Gaussian function [33]. As shown in Figure 2b, the SPDs of the monochromatic red and green LEDs, along with their corresponding Gaussian fitting curves, are presented. The high values of coefficient of determination (R²) indicate the rationality of the Gaussian fittings. Thus, the SPDs of the monochromatic red and green LEDs can be expressed in the mathematical form as follows:

$$\left\{\begin{array}{c}{P}_R(\lambda )=P_{opt,R}{\displaystyle \frac{1}{{\sigma }_R\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,R}{)}^2}{{{\sigma }_R}^2}}\right]\\ P_G(\lambda )=P_{opt,G}{\displaystyle \frac{1}{{\sigma }_G\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,G}{)}^2}{{{\sigma }_G}^2}}\right]\end{array}\right.$$

(6)

where P_opt,R and P_opt,G are the same as defined in Equation (1); λ_peak,R and σ_R are, respectively, the peak wavelength and the rated coefficient of full-width at half-maximum (FWHM) of the monochromatic red LED; λ_peak,G and σ_G are, respectively, the peak wavelength and the rated coefficient of FWHM of the monochromatic green LED.

Combining Equations (2), (5) and (6), the S/P of the compensated white LED system can be expressed as

$$\begin{gathered} S/P={\displaystyle \frac{K_s{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+P_{opt,R}\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda –{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+P_{opt,G}\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda –{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}{V}^{\prime }\left(\lambda \right)d\lambda }{K_m{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+P_{opt,R}\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda –{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+P_{opt,G}\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda –{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}V\left(\lambda \right)d\lambda }} \\ ={\displaystyle \frac{K_s{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+\left(k_{R1}+k_{R2}{I}_R\right)\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+(k_{G1}+k_{G2}{I}_G)\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda –{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}{V}^{\prime }\left(\lambda \right)d\lambda }{K_m{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+\left(k_{R1}+k_{R2}{I}_R\right)\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda –{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+(k_{G1}+k_{G2}{I}_G)\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}V\left(\lambda \right)d\lambda }} \end{gathered}$$

(7)

Since the peak wavelength and the corresponding FWHM of a monochromatic LED remain essentially unchanged with the driving current, parameters λ_peak,R, λ_peak,G, σ_R and σ_G can be treated as constants for simplified analysis [34]. Additionally, the SPD of the PC white LED, i.e., P_W(λ), is also fixed due to the stable operating conditions of the PC white LED. Therefore, the expression of S/P in Equation (7) can be further simplified as

$$S/P\left(I_R,I_G\right)={\displaystyle \frac{K_s}{K_m}}{\displaystyle \frac{S_{WRG}+S_R{I}_R+S_G{I}_G}{P_{WRG}+P_R{I}_R+P_G{I}_G}}$$

(8)

In Equation (8), the parameters S_WRG, P_WRG, S_R, P_R, S_G and P_G are introduced to simplify the expression of the S/P model, and their specific definitions are provided as follows:

$$\left\{\begin{array}{c}\begin{array}{c}{S}_{WRG}={\int }_{380}^{780}\left\{P_W\left(\lambda \right)+k_{R1}{\displaystyle \frac{1}{{\sigma }_R\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,R}{)}^2}{{{\sigma }_R}^2}}\right]+k_{G1}{\displaystyle \frac{1}{{\sigma }_G\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,G}{)}^2}{{{\sigma }_G}^2}}\right]\right\}{V}^{\prime }\left(\lambda \right)d\lambda \\ P_{WRG}={\int }_{380}^{780}\left\{P_W\left(\lambda \right)+k_{R1}{\displaystyle \frac{1}{{\sigma \sigma }_R\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,R}{)}^2}{{{\sigma }_R}^2}}\right]+k_{G1}{\displaystyle \frac{1}{{\sigma }_G\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,G}{)}^2}{{{\sigma }_G}^2}}\right]\right\}V\left(\lambda \right)d\lambda \end{array}\\ \begin{array}{c}{S}_R={\int }_{380}^{780}{k}_{R2}{\displaystyle \frac{1}{{\sigma }_R\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,R}{)}^2}{{{\sigma }_R}^2}}\right]V^{\prime }\left(\lambda \right)d\lambda \\ S_G={\int }_{380}^{780}{k}_{G2}{\displaystyle \frac{1}{{\sigma }_G\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,G}{)}^2}{{{\sigma }_G}^2}}\right]V^{\prime }\left(\lambda \right)d\lambda \\ P_R={\int }_{380}^{780}{k}_{R2}{\displaystyle \frac{1}{{\sigma }_R\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,R}{)}^2}{{{\sigma }_R}^2}}\right]V\left(\lambda \right)d\lambda \end{array}\\ P_G={\int }_{380}^{780}{k}_{G2}{\displaystyle \frac{1}{{\sigma }_G\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,G}{)}^2}{{{\sigma }_G}^2}}\right]V\left(\lambda \right)d\lambda \end{array}\right.$$

(9)

Given the constancy of these foundational parameters (k_R1, k_R2, k_G1, k_G2, λ_peak,R, σ_R, λ_peak,G, σ_G and P_W(λ)), the derived parameters S_WRG, P_WRG, S_R, P_R, S_G and P_G can also be regarded as constants.

Equation (8) indicates that the S/P of the mixed spectra emitted from the LED system based on spectral compensation is also determined by the driving currents of the monochromatic red and green LEDs, denoted as I_R and I_G.

2.3. Blue Light Hazard (BLH) Efficacy

BLH generally refers to the potential photochemical risk to the retinal tissues of human eyes caused by the radiation exposure in the wavelength range of 400–500 nm [35]. For a light source, the potential for retinal BLH is evaluated via the BLH efficacy, which is defined as the ratio of the blue-light-weighted radiometric quantity to the photometric quantity [36], as follows:

$$BLH={\displaystyle \frac{{\int }_{380}^{780}P\left(\lambda \right)B\left(\lambda \right)d\lambda }{K_m{\int }_{380}^{780}P\left(\lambda \right)V\left(\lambda \right)d\lambda }}$$

(10)

where BLH denotes the BLH efficacy, V(λ) is the spectral sensitivity functions under photopic vision, B(λ) represents the BLH spectral weighting function characterized by a wavelength peak approximately in the range of 435–440 nm. Additionally, the constant parameter K_m is consistent with that in Equation (4), i.e., the maximum spectral luminous efficacy for photopic vision, with a standard value of 683 lm/W.

Substituting P(λ) = P_W(λ) + P_R(λ) + P_G(λ) into Equation (10), the corresponding BLH can be rewritten as

$$BLH={\displaystyle \frac{{\int }_{380}^{780}\left[P_W\left(\lambda \right)+P_R\left(\lambda \right)+P_G\left(\lambda \right)\right]B\left(\lambda \right)d\lambda }{K_m{\int }_{380}^{780}\left[P_W\left(\lambda \right)+P_R\left(\lambda \right)+P_G\left(\lambda \right)\right]V\left(\lambda \right)d\lambda }}$$

(11)

Now, combining Equations (2), (6) and (11), the BLH of the compensated white LED system is expanded as

$$\begin{gathered} BLH={\displaystyle \frac{{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+P_{opt,R}\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+P_{opt,G}\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}B\left(\lambda \right)d\lambda }{K_m{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+P_{opt,R}\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+P_{opt,G}\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}V\left(\lambda \right)d\lambda }} \\ ={\displaystyle \frac{{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+(k_{R1}+k_{R2}{I}_R)\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+(k_{G1}+k_{G2}{I}_G)\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}B\left(\lambda \right)d\lambda }{K_m{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+(k_{R1}+k_{R2}{I}_R)\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+(k_{G1}+k_{G2}{I}_G)\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}V\left(\lambda \right)d\lambda }} \end{gathered}$$

(12)

Similar to the derivation of S/P model, the parameters k_R1, k_R2, k_G1, k_G2, λ_peak,R, λ_peak,G, σ_R, σ_G and P_W(λ) are treated as constants; thus, the BLH efficacy in Equation (12) can be further simplified to an expression with the driving currents I_R and I_G as variables, that is

$$BLH\left(I_R,I_G\right)={\displaystyle \frac{1}{K_m}}{\displaystyle \frac{B_{WRG}+B_R{I}_R+B_G{I}_G}{P_{WRG}+P_R{I}_R+P_G{I}_G}}$$

(13)

where the parameters P_WRG, P_R and P_G are consistent with their definitions in Equation (9), and the derived constants parameters B_WRG, B_R and B_G are given as follows:

$$\left\{\begin{array}{c}{B}_{WRG}={\int }_{380}^{780}\left\{P_W\left(\lambda \right)+k_{R1}{\displaystyle \frac{1}{{\sigma }_R\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,R}{)}^2}{{{\sigma }_R}^2}}\right]+k_{G1}{\displaystyle \frac{1}{{\sigma }_G\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,G}{)}^2}{{{\sigma }_G}^2}}\right]\right\}B\left(\lambda \right)d\lambda \\ B_R={\int }_{380}^{780}{k}_{R2}{\displaystyle \frac{1}{{\sigma }_R\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,R}{)}^2}{{{\sigma }_R}^2}}\right]B\left(\lambda \right)d\lambda \\ B_G={\int }_{380}^{780}{k}_{G2}{\displaystyle \frac{1}{{\sigma }_G\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,G}{)}^2}{{{\sigma }_G}^2}}\right]B\left(\lambda \right)d\lambda \end{array}\right.$$

(14)

Equation (13) indicates that the BLH efficacy of the mixed spectra from the compensated white LED system can also be determined using the driving currents I_R and I_G of the monochromatic red and green LEDs.

2.4. Circadian Action Factor (CAF)

As previously described, light plays a critical role in regulating human physiological functions. To quantify the specific contribution of light sources to this regulatory process, the CAF is defined as [17]

$$CAF={\displaystyle \frac{{\int }_{380}^{780}P\left(\lambda \right)C\left(\lambda \right)d\lambda }{{\int }_{380}^{780}P\left(\lambda \right)V\left(\lambda \right)d\lambda }}$$

(15)

where V(λ) is also the spectral sensitivity functions under photopic vision, and C(λ) is the circadian spectral sensitivity function with a peak wavelength of 450 nm.

By substituting P(λ) = P_W(λ) + P_R(λ) + P_G(λ) into Equation (15), the corresponding CAF can be derived as

$$CAF={\displaystyle \frac{{\int }_{380}^{780}\left[P_W\left(\lambda \right)+P_R\left(\lambda \right)+P_G\left(\lambda \right)\right]C\left(\lambda \right)d\lambda }{{\int }_{380}^{780}\left[P_W\left(\lambda \right)+P_R\left(\lambda \right)+P_G\left(\lambda \right)\right]V\left(\lambda \right)d\lambda }}$$

(16)

Similarly, based on the combination of Equations (2), (6) and (16), the CAF of the compensated white LED system is expressed in an expanded form as

$$\begin{gathered} CAF={\displaystyle \frac{{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+P_{opt,R}\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+P_{opt,G}\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}C\left(\lambda \right)d\lambda }{{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+P_{opt,R}\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+P_{opt,G}\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}V\left(\lambda \right)d\lambda }} \\ ={\displaystyle \frac{{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+(k_{R1}+k_{R2}{I}_R)\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+(k_{G1}+k_{G2}{I}_G)\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}C\left(\lambda \right)d\lambda }{{\int }_{380}^{780}\left\{P_W\left(\lambda \right)+(k_{R1}+k_{R2}{I}_R)\frac{1}{{\sigma }_R\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,R}\right)}^2}{{{\sigma }_R}^2}\right]+(k_{G1}+k_{G2}{I}_G)\frac{1}{{\sigma }_G\sqrt{2\pi }}\mathit{exp}\left[-2\frac{{\left(\lambda -{\lambda }_{peak,G}\right)}^2}{{{\sigma }_G}^2}\right]\right\}V\left(\lambda \right)d\lambda }} \end{gathered}$$

(17)

As noted in the derivations of S/P and BLH models, the parameters k_R1, k_R2, k_G1, k_G2, λ_peak,R, σ_R, λ_peak,G, σ_G and the P_W(λ) are all treated as constants. Therefore, Equation (17) corresponding to the CAF can also be expressed as a mathematical formula with the driving currents I_R and I_G as variables, as follows:

$$CAF\left(I_R,I_G\right)={\displaystyle \frac{C_{WRG}+C_R{I}_R+C_G{I}_G}{P_{WRG}+P_R{I}_R+P_G{I}_G}}$$

(18)

where the constant parameters P_WRG, P_R and P_G are also consistent with those defined in Equation (9), while C_WRG, C_R and C_G are additional derived constant parameters, whose specific definitions are provided as follows:

$$\left\{\begin{array}{c}{C}_{WRG}={\int }_{380}^{780}\left\{P_W\left(\lambda \right)+k_{R1}{\displaystyle \frac{1}{{\sigma }_R\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,R}{)}^2}{{{\sigma }_R}^2}}\right]+k_{G1}{\displaystyle \frac{1}{{\sigma }_G\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,G}{)}^2}{{{\sigma }_G}^2}}\right]\right\}C\left(\lambda \right)d\lambda \\ C_R={\int }_{380}^{780}{k}_{R2}{\displaystyle \frac{1}{{\sigma }_R\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,R}{)}^2}{{{\sigma }_R}^2}}\right]C\left(\lambda \right)d\lambda \\ C_G={\int }_{380}^{780}{k}_{G2}{\displaystyle \frac{1}{{\sigma }_G\sqrt{2\pi }}}exp\left[-2{\displaystyle \frac{(\lambda -{\lambda }_{peak,G}{)}^2}{{{\sigma }_G}^2}}\right]C\left(\lambda \right)d\lambda \end{array}\right.$$

(19)

Based on Equation (18), the CAF of the white LED system with compensation of red and green spectra can also be obtained from the driving conditions of the monochromatic red and green LEDs.

From the discussions on total optical power, S/P, BLH and CAF, the following crucial aspects can be summarized:

(1)

For the compensated white LED systems studied, the spectral power distribution is determined by the operating states of the PC white LED, the monochromatic red LED, and the monochromatic green LED. The PC white LED serves as the core component, operating in a steady state to provide the necessary luminous flux for lighting. Meanwhile, the operating states of the monochromatic red and green LEDs are adjustable considering their stable monochromaticity.

(2)

The established models of total optical power, S/P, BLH and CAF can be expressed as functions of the driving currents I_R and I_G corresponding to the monochromatic red LED and monochromatic green LED, respectively. As a consequence, by controlling these two LEDs’ electrical driving conditions, the spectral structure of the mixed white light can be altered, enabling the regulation of visual performance and photobiological effects.

(3)

The S/P, BLH and CAF of a light source primarily depend on the spectral power distribution. The approach of spectral compensation can alter the intensity of light stimuli on different types of photoreceptor cells in the human retina, thereby influencing visual performance and photobiological effects.

(4)

Although the models of S/P, BLH and CAF exhibit non-linearity, lighting design engineers can still achieve a rapid estimation of these factors from the driving currents of the monochromatic red and green LEDs, and thus effectively understand the visual perception and photobiological effects of the illumination environments. Furthermore, these models provide valuable reference frameworks to develop lighting solutions that are more human-centered and health-oriented.

3. Verifications and Discussion

3.1. Measuring Instruments and Operating Conditions

The proposed models for the visual performance and photobiological effects are validated using two LED systems: CW-R-G LED system and WW-R-G LED system, whose structural schematic can be seen from Figure 1. It is worth mentioning that for the monochromatic green LED, its λ_peak,G and σ_G are 524 nm and 39.1, respectively; for the monochromatic red LED, its λ_peak,R and σ_R are 628 nm and 25.9, respectively; for the PC warm white LED in the WW-R-G LED system, it has a low CCT of 3600 K; for the PC cool white LED in the CW-R-G LED system, it has a high CCT of 6000 K.

The experimental setup, as shown in Figure 3, incorporates several key components, including: (1) an Everfine HAAS-2000 spectroradiometer system equipped with an integrating sphere photometer, utilized for conducting optical measurements; (2) a Peltier-cooled fixture CL-200, employed as an efficient temperature-controlled heatsink to guarantee optimal thermal management; (3) two external power supplies, i.e., the Rigol programmable power supplies, applied to supply stable and precisely adjustable electrical currents to the LEDs; (4) control platform implemented by a workstation, used for operation control and data analysis via specific software.

During the optical measurement process, the heatsink is maintained at a constant temperature of 25 °C. The PC white LEDs in the systems operate at a fixed current of 300 mA (30% of their 1000 mA rated current), with a corresponding optical power of approximately 500 mW. This enables them to operate in a “highly efficient unsaturated region”, characterized by uniform current distribution, excellent power–current linearity, and stable electro-optical efficiency. As compensation light sources, the monochromatic red and green LEDs have driving currents that do not exceed half of that of the main light source (i.e., the PC white LED); accordingly, their adjustable current range is set to 20–140 mA, with an increment interval of 20 mA. Furthermore, to ensure the stability and reliability of the data, all optical measurements are conducted after a 20-min period of stable operation.

3.2. Result Analysis

According to the proposed models, the total optical power P_opt, S/P, BLH and CAF of the white LED systems based on spectral compensation are calculated with electrical input values. The calculated results and their corresponding experimental measurements are presented in Figure 4, Figure 5, Figure 6 and Figure 7, respectively of the CW-R-G LED system and WW-R-G LED system. Notably, a high degree of agreement can be observed between the measured and calculated results, both in terms of numerical values and variation trends.

As shown in Figure 4, the total optical power P_opt of the compensated white LED system exhibits a consistent increasing trend with the rise in either the driving current I_R (of the monochromatic red LED) or I_G (of the monochromatic green LED). Moreover, the relationship between P_opt and each of these two currents maintains excellent linearity, which is consistent with the optical power model shown in Equation (3).

As illustrated in Figure 5, at a fixed current I_R, the S/P value increases with the intensity of current I_G, which means that it becomes more appropriate for night-time lighting or low-illumination scenes to offer enhanced visual perception and effective visual guidance. Conversely, when the current I_G is kept constant, the S/P value decreases as the intensity of current I_R rises. This suggests that the illumination can generate sufficiently bright visual luminance, enabling users to clearly distinguish object details and to improve work efficiency and visual experience. This makes it particularly suitable for applications such as library reading areas, office workspaces, and shopping mall displays.

It can be seen from Figure 6 that when the current I_R or current I_G rises, the BLH efficacy is showing a downward trend. This indicates consistent relationships between these two input currents and the BLH efficacy. Specifically, any increment in either current I_R or current I_G can alleviate the degree of blue light hazard. In other words, compensation of red and green spectra is beneficial for reducing blue light hazard of the PC white LED, making the lighting environment healthier and less likely to cause retinal damage.

As shown in Figure 7, the relationships between the input currents (I_R and I_G) and the CAF are similar to those between the currents and the BLH efficacy. That is, an increase in either current I_R or current I_G will lead to a decrease in the CAF, indicating that the impact on human circadian rhythm becomes weakened. Specifically, for the CW-R-G LED system, the CAF values generally fall within the range of 0.35–0.65. This indicates that it can effectively suppress the secretion of melatonin, thereby keeping people awake and alert. Consequently, it is suitable for situations such as in offices or libraries, as it is conducive to improving concentration and efficiency. For the WW-R-G LED system, its CAF mainly ranges from 0.15–0.35, and the circadian rhythm stimulation associated with light is beneficial for the normal secretion of melatonin. Hence, it is recommended for lounges or bedrooms, helping to relieve tension and create a relaxing and soothing atmosphere.

Seen from Figure 4 and Figure 5, the linearity of both the total optical power and S/P with respect to the driving currents (I_R for the monochromatic red LED and I_G for the monochromatic green LED) is generally excellent. In contrast, as shown in Figure 6 and Figure 7, respectively, the linearity between the BLH and the current, as well as that between the CAF and the current, is relatively weak. Moreover, as the current I_G increases, the descending gradients of both BLH and CAF exhibit a diminishing trend. Combined with the electroluminescence mechanism of LEDs, this phenomenon is potentially related to the fact that, as the driving electrical power increases, the emission spectra of GaN-based green LEDs shift to the short-wave band [37,38]. This is the phenomenon known as the blue shift [39]. In addition, all the curves presented in Figure 4, Figure 5, Figure 6 and Figure 7 are vertically evenly spaced. It indicates a high degree of linearity, as values of total optical power, S/P, BLH, and CAF change uniformly with the increasing current I_R. This also provides support for the fact that no obvious spectral shift occurs in monochromatic red LEDs. The emission spectra of the monochromatic red and green LEDs versus their driving currents are illustrated in Figure 8. Here, the emission spectra of each individual LED under different currents are normalized, thus allowing for a more distinct observation of the spectral shift.

Furthermore, the absolute and relative errors of the results are also calculated as quantitative supplements, with the corresponding maximum and mean values presented in

Table 1. Errors analysis of S/P, BLH and CAF.

		Maximum Absolute Error	Mean Absolute Error	Maximum Relative Error	Mean Relative Error
CW-R-G LED system	Optical power	18.71 mW	4.90 mW	3.72%	0.79%
	S/P	0.04374	0.01107	2.16%	0.58%
	BLH	0.24606 × 10−4	0.05845 × 10−4	3.52%	0.87%
	CAF	0.02069	0.00383	3.26%	0.79%
WW-R-G LED system	Optical power	24.64 mW	5.82 mW	5.89%	1.01%
	S/P	0.05386 × 10−4	0.02068 × 10−4	4.41%	1.33%
	BLH	0.11808	0.02539	3.04%	0.87
	CAF	0.00385	0.00152	1.75%	0.68%

. Notably, all errors associated with total optical power, S/P, BLH, and CAF are sufficiently small: the maximum relative errors do not exceed 5.9%, and the mean relative errors are within 1.4%.

The high consistency between the calculated and measured results (as illustrated in the figures) and the error analysis collectively validate the accuracy and reliability of the proposed models, thereby demonstrating their effectiveness in predicting the visual performance and photobiological effects of compensated white LED systems.

3.3. Performance Comparison

The prominent advantage of the proposed compensated white LED systems is that they can simultaneously achieve tunable CCT while ensuring a sufficiently high CRI. Here, a comparison with a dual-color white LED system has been discussed, as follows.

For the CW-R-G system, when the red and green LEDs operate with changing currents, it achieves a tunable CCT range of 3413–5311 K and a satisfactory CRI range of 80–89.3, accompanied by a gradually increasing light purity from 3.7 to 27.1. Compared with the uncompensated PC cool white LED (CRI = 75.9), the maximum CRI improvement rate reaches 17.65% and the light purity increases by more than 6 times. Correspondingly, for the WW-R-G system, it obtains a tunable CCT range of 2343–4521 K and a satisfying CRI range of 80.2–95.3, accompanied by an excellent light purity gradually increasing from 41.7 to 60.9. In comparison with the uncompensated PC warm white LED (CRI = 77.8), the maximum CRI improvement rate is as high as 22.49% and the light purity has increased by 46%.

The dual-color white LED system for comparison is integrated by two PC white LEDs, one with a high CCT of 7600 K, a CRI of 79.6 and a light purity of 14; the other with a low CCT of 2700 K, a CRI of 82.3 and a light purity of 55. When integrated into the dual-color white LED system, the CCT of the mixed spectra can be adjusted from 2700 K to 7600 K, while the CRI ranges from 79.8 to 84.4, the maximum CRI improvement rate is only 6.03%. Regarding the light purity of the integrated system, no significant correlational relationship was observed between its values and the spectral intensity of the two PC white LEDs. Specifically, as the spectral intensity of the PC white LEDs varied, the light purity sometimes increases but sometimes decreases, basically falls within the range of 3.5–44.

Since the CCT range considered ideal for general lighting typically falls between 2300 K and 5300 K, the compensated white LED systems are able to deliver a more appropriate lighting environment when comprehensively considering both the tunable CCT and improved CRI.

4. Conclusions

In this work, the visual performance and photobiological effects of white LED systems based on spectral compensation are investigated, and models of S/P, BLH and CAF are proposed which link up the spectral power distribution and spectral efficiency functions of the human visual system. The proposed models mathematically explain the intuitive influence of electrical driving conditions of LEDs on the total optical power, S/P, BLH and CAF. In the experimental verification, excellent agreement is exhibited between the calculations and their corresponding measurements, indicating the rationality and accuracy of the models and their establishment. These proposed models are expected to serve as powerful tools for comprehending the visual perception and photobiological effects of illumination, to provide guidance for the design of LED lighting products with more optimized spectral distributions, and to reduce the negative impact on human health and to improve the overall lighting experience.

The proposed models are primarily designed for the compensated white LED system, where the PC white LED serves as the core component and operates in a steady state to provide the basic luminous flux required for lighting, and the monochromatic red and green LEDs function as spectral compensators to enable adjustment of spectral structure and improvement of optical performance. More generally, the application scope of these models can be further expanded to any multi-LED system where one or several LEDs act as stable core light sources (maintaining a fixed operating state) and the remaining multiple monochromatic LEDs serve as adjustable spectral compensators (allowing regulation of electrical driving state). For such systems, the optical power, S/P, BLH, and CAF models will exhibit forms similar to those proposed in this paper. Though certain parameters of the models will vary depending on the specific composition of the system, acquiring these parameters poses little difficulty for engineers engaged in LED system design.