2.1. LED Emission Spectra

A systematic BLH analysis of LED parameter variations can be conducted either by determining experimentally optical measurands like (ir-) radiance or by a calculative approach with the semiconductor’s radiation emission, hereinafter simply referred to as signal S(λ), described mathematically. Based on the fundamental physics of semiconductors, S(λ) can be approximated by a Gaussian function [21],

with peak intensity S0, wavelength λ, peak wavelength λ0, and spectral bandwidth Δλ0. For simplicity, S(λ) is not related to a specific physical quantity so that S0 is dimensionless. Other mathematical descriptions can also be found in literature, e.g., using a sum of several Gaussian or cosine-power functions [19,20]. Equation (1) is not normalized to an area of 1; thus, the desired spectral bandwidth Δλ=1.178 Δλ0 (derived empirically) must be corrected. One example for an LED signal according to Equation (1) is presented in Figure 1. The three Gaussian parameters are written as triple λ0|Δλ0|S0 that is 445|25|1 for the depicted blue-LED light emission. Hereinafter, such single emission line LEDs will be referred to as color LEDs although there is no color perception of the human visual apparatus in the ultraviolet spectral region.

Exemplary signal S(λ) for a color and a phosphor-conversion white-light emitting diode (pc-LED) according to Equation (1). The blue-LED signal (dashed line), centered at the peak wavelength λ0= 445 nm with a spectral bandwidth of Δλ0= 25 nm, is normalized to its maximum, S0. The yellowish-green phosphor emission with parameter triple λph|Δλph|Sph= 560|125|0.5 is given as dash-dotted line. The addition of both Gaussian curves, subsequently peak normalized, represents a pc-LED (solid line) with x= 0.30, y= 0.31, and Tcp= 7396 K.

Approximating the YAG phosphor fluorescence by a second Gaussian function according to Equation (1) with parameter triple λph|Δλph|Sph allows a mathematical description of some pc-LED emission spectra. Adding both Gaussian functions, λ0|Δλ0|S0+λph|Δλph|Sph, and subsequently normalizing the sum results in a characteristic emission spectrum, see Figure 1. It has to be noted that neither a temperature-induced asymmetric line broadening nor a long-wavelength tailoring of the phosphor’s light emission can be considered by Equation (1). The latter is negligible for BLH due to the minor (≤0.001) relative spectral effectiveness of B(λ) for λ≥ 600 nm, but it can have a large effect on the luminous signal, Sv, weighted by the spectral luminous efficiency for photopic vision, V(λ).