Artificial Intelligence in Light-Source Design

The advent of light-emitting diode (LED) light sources has led to a new freedom in the design of light-source spectra, and it is now possible to optimise for different source performance parameters, which is the principal aim of the authors ’ work. LEDs and lasers are real or potential light sources, and are inherently monochromatic, that is, narrow-band sources, with typical optical bandwidths in the range 20 – 40 nm (nanometres) for LEDs and 1 – 5 nm for diode lasers. Mixtures of three or more can be used to produce nominally white light of the type acceptable for general purpose lighting. It is a characteristic of all types of sources that there is a trade-off between good colour properties and high efficiencies, and the methods described here are directed towards an optimum combination of such parameters. This chapter will explain the use of differential evolution (DE) as a highly effective heuristic approach to optimisation, and proceeds to explain the structure and operation of a DE algorithm designed as an optimisation tool for such purposes.


Introduction
It has been stated by some authors that white LEDs are the eco-friendly light sources for the twenty-first century [1]. White light in LEDs can be produced either (i) by using blue (or violet) light to irradiate a phosphor that emits yellow light [2,3] or (ii) by combining the outputs of a group of monochromatic LEDs [1,4,5]. In approach (i) the blue or violet is combined with the yellow to produce white light. We have adopted approach (ii) in which three (or, preferably, four) primary colours are mixed in appropriate proportions to produce high-quality white light with higher efficiency than approach (i) since there is no loss of energy as in the down-conversion in the phosphor [5,6]. Both approaches aim to design LED-based white-light sources characterised by high colour rendering and fidelity properties combined with high luminous efficacy. It is an established fact in light-source design that these two qualities are generally contravariant, and that both depend on the emitted spectrum of the source. We have found that fine-tuning of the intensities of the individual sources, as well as the selection of their individual peak wavelengths, enables us to define the best balance of the significant spectral properties of the mixture. Our adopted optimisation technique implements a trial-and-error search algorithm within an n-dimensional space {I 1 , …, I i , …, I n } until an acceptable spectral power distribution (SPD) is found for the white-light mixture [5,7,8].
In this chapter, we will describe an approach to the design of the optimum LED mixture using a simple global optimisation algorithm based on differential evolution (DE) as proposed by Storn and Price [9]. We have optimised the SPDs of mixtures of real and simulated LEDs and diode lasers. For this purpose, a MATLAB program was developed that optimises the SPD of an arbitrary mixture of individual SPDs. Specific versions of this program were developed to implement several different techniques for defining (and calculating) the colour properties of sources.
This chapter proceeds as follows. In Section 2, the principles of the DE algorithm are described in detail. In Section 3, the theory of colour rendering and luminous efficacy is given, along with other relevant lighting terms and their usage. Section 4 presents a selection of key experimental results and comparisons and, finally, we state our conclusions in Section 5.

Differential evolution
Differential evolution is a powerful population-based evolutionary algorithm suitable for the optimisation of real-value multi-modal non-linear and nondifferentiable objective functions f o (x 1 , x 2 , ..., x n ) [9,10]. DE is simple and has proven to be powerful in solving a number of benchmark problems [11,12].
The search for an optimal solution starts with a population of P randomly created solution vectors {v 1 , v 2 , ..., v P }. This population is maintained constant during the optimisation process, during which the solution vectors (i.e., candidate SPDs) undergo mutation, crossover, evaluation and selection over a number of generations G. The best choices for both the population size P and the number of generations G depend on the problem to be optimised. Storn and Price suggest P ∈ [5n, 10n], where n is the dimension of the objective function, but P must be ≥4 to provide sufficient mutually different solution vectors for the algorithm to function properly [10].
The operation of the algorithm is controlled by the use of a fitness function f fit designed to discriminate between solutions (SPDs here) with 'good' or 'poor' properties.
Mutation is the process of creating a new offspring vector u i,G + 1 by adding the weighted difference between two randomly chosen solution vectors, v r2,G and v r3,G , to a randomly chosen v r1,G [10]: where F is a mutation weight ∈ [0,2] and random indices r 1 , r 2 , r 3 ∈ [0, PÀ1] are chosen to be different from the index i. The mutation process is illustrated in Figure 1.
The offspring solution vectors u i,G + 1 undergo crossover which ensures that the offspring vectors w ji,G + 1 differ from their parents [10]: where the terms and parameters are defined as follows: i ∈ {1,2,..., P} is a randomly chosen integer, j ∈ [0, 1) is a randomly chosen real value, CR ∈ [0, 1] is a crossover constant influencing the number of elements to be exchanged. An example of the crossover process is shown in Figure 2.
The process compares the offspring vectors {w 1,G + 1 ,..., w P,G + 1 } against their parent vectors {v 1,G ,..., v P,G }. If the offspring w i,G + 1 has a better fitness function f fit than its parent v i,G , then it becomes a member of the next generation, G + 1. If not, the v i,G vector is retained. Hence, only the fitter offspring become members of the G + 1 generation.
Proposals have been made for several variants of the original DE algorithm [10]. The most significant differences are in the creation of new solution vectors where: 1. the best solution vector from the current generation v best,G is mutated rather than v r1,G ; 2. more than two difference vectors are used in mutation; and 3. different crossover schemes are employed.
Our work is based on the original DE, and is described in pseudo-code given below: Require

Light-source properties
Our purpose in this chapter is to present an approach to intelligent spectral design for any white-light source, aiming to achieve an optimum combination of luminous efficacy and colour rendering which, as previously noted, are contravariant characteristics of the SPD. We next introduce a brief outline of these, and other important, source properties.

Colour rendering index
Colour rendering has been defined by the Commission Internationale de l'Éclairage, International Commission on Illumination (CIE) who have published recommendations for the method of calculation of their colour rendering index (CRI) [13] based on a knowledge of the light-source spectrum. It represents an evaluation of the average colour shift of eight defined moderate-chroma colour samples when compared under the test source and a reference source having the same correlated colour temperature (CCT). 1 The system includes 14 test colours in total, and the additional 6 comprise 4 highly saturated colours (red, yellow, green, and blue) plus samples representing skin and foliage colours, respectively. As of the time of writing, this is the internationally agreed method. Note that CRI and associated technology have also been covered in [14]. The two most widely quoted colour rendering terms are: R a -the general colour rendering index, based on the colour shifts of the eight principal test colour samples; and R 9 -the 'special' (individual) index for the highly saturated red colour (sample 9).
In our optimisation work (see Section 4.1), we also made use of several derived indices symbolised as R b , R c and R min , based, respectively, on: the 6 additional test colours; the full set of 14 test colours and the minimum individual value from the R c set.

Colour fidelity index
Some dissatisfaction with the CIE method has arisen since the widespread adoption of LED lighting. As a consequence, the Illumination Engineering Society of North America (IES) has adopted a recommended method (TM-   [15,16] which recommends two new indices (R f and R g ) for the classification of the colour properties of light sources. The underpinning research leading to the development of TM-30-15 [16][17][18] identified several weaknesses in the CIE's earlier CRI method [13], claiming that it does not adequately sample wavelength space and hence tends to over-estimate colour performance.
The method is also based on the colour-shift concept, but now using a set of 99 test colours considered to provide uniformity of both wavelength sampling and colour-space sampling. In addition, it uses a more modern colour-difference calculation technique, CAM02UCS [19] which is a development of the basic CIECAM02 colour appearance space [20].
The new index R f gives an overall assessment of colour fidelity, while gamut index R g indicates the relative magnitudes of colour shifts for sample colours in different regions of colour space. Also available is the skin colour index, R f skin , which is an average of two specific sample-colour indices, selected as representative of human skin. In our optimisations, we also called up the minimum value of R f , symbolised as R f min , from the full set of 99 individual indices R f i , as well as the referenced colour sample number i min (in the set i 1 to i 99 ).

Colour quality scale
This was a precursor to TM-30-15, first proposed by Davis and Ohno of NIST (USA) [21]. The following serves as a brief introduction for the purpose of the present discussion. Again using the previously mentioned colour-shift concept, the CQS metric employs 15 saturated test colour samples, on the premise that certain light sources may render saturated colours more poorly than the de-saturated colours of the CIE's CRI method. The chromatic differences are calculated using the CIE 1976 (CIELAB) colour model [22].
The full calculation procedure has also been explained in [23]. It employs multiple steps, several of which are non-linear, resulting in the general CQS index, Q a . As with the previous two cases, the 'special CQS' (Q i ) for each test colour sample may be calculated for a more thorough investigation of a test source. We have used the minimum value of Q i (designated Q min ) from the set of 15 Q i values, as an optimisation parameter.

Luminous efficacy of radiation
The luminous efficacy of the radiation (LER) of a light source assesses the 'lighting content' of the spectrum by comparing the visible light output (in lumens) to the total radiant output (in watts) as in Eq. 3.
where K m is the maximum luminous efficacy of radiation (≈683 lumen per watt), S(λ) is the spectral distribution of the light source, and V(λ) is the CIE spectral sensitivity function for human photopic vision [24]. The LER is an important determinant of the overall economy of a light source since the overall luminous efficacy is given by the product of LER with the energy conversion efficiency of the particular light source.

Correlated colour temperature
From the perspective of the lighting system designer, the correlated colour temperature (CCT) is the key feature in the selection of a light source since the CCT serves as an indicator of, not only the colour of the source, but also, the 'atmosphere' it will create.
The CCT is defined in [25], and its significance is that it describes the chromaticity of the source (which must be close to the Planckian locus) in the CIE (u, v) chromaticity diagram. 2 Note that it is possible for many different spectral power distributions (SPDs) to have the same CCT; and that CCT is not essentially linked with colour rendering, quality or fidelity. 2 In CIE documentation, this is now replaced by the u 0 ; 2 coordinates.

Optimisations
We here present, in outline form, the results of optimised spectral designs achieved using DE. Interested readers are referred to previous publications [23,[26][27][28][29] for full details.
It is noted at the outset that it is, in principle, possible to create white-light mixtures by the use of as few as two or three wavelengths. Our early experiments showed that these practically always lead to suboptimal mixtures in the sense that either one or both of LER and X c are unacceptably low (X c is here used to denote any one of the colour metrics mentioned in Section 3). Further experiments indicated that mixtures of five, six or seven wavelengths gave little or no practical advantage over 4-band mixtures, while adding to the complexity (and possible unreliability) of the light source. For the sake of brevity, therefore, the following descriptions are confined solely to results for 4-band mixing.
Two types of optimisation conditions are described: (i) constrained, in which the optimiser is presented with a set of known (e.g., commercial) monochromatic spectra, and is required to find the best available mixtures in terms of defined criteria and (ii) unconstrained, when the program is given mathematical descriptions for the shapes of potential monochromatic spectra, which it then proceeds to mix and optimise as above.
In the optimisations described below, all SPDs for LEDs were computed at 5-nm intervals, and for lasers at 1 nm.

Optimisation of LED mixtures for LER and CRI (CIE R a )
This investigation [26] explored the optimisation of white-light mixtures of seven LEDs (four at a time) from the Lumileds Luxeon™ range [30] and it is therefore classed as constrained. The objective here was to achieve the best available combination of R a with LER, and the fitness function for DE was defined as in Eq. 4: where R a = CIE general colour rendering index (based on 8 medium-chroma test colours); R b = similar figure of merit based on the 6 additional test colours in the CIE method; R c = similar figure of merit based on all 14 test colours in the CIE method; η rad = the value of the LER (lumen per watt figure); R min = the lowest value of R i in the set of 14 individual values; a, b, c, d and e are user-selectable weights controlling the influence of R a , R b , R c , η rad and R min on the optimisation process.
Experimentation found that f fit converged around (or before) the 1000th generation; hence, the number of generations G was set to 1000 for fast and accurate convergence of the DE. Note that a too-small G can mean that the process has insufficient opportunity to search for effective solution vectors whereas, for excessive values of G, the optimisation process is unnecessarily slowed down without improving the optimisation result.
The other DE parameters that may influence the operation of the algorithm are P, F, and CR, all of which need to be set prior to any run. We found by experimentation that the values of these parameters had only a minor influence on the optimisation results; and the choice of suitable values for good optimisation was straightforward. We quickly settled on a value of P = 50, and we set F and CR according to the suggestions in [10], that is, F = 0.5 and CR = 0.1.
Further experimentation investigated the effects of different values for the weighting factors for f fit , and some indicative results are shown in Table 1. The relative spectral power distribution (SPD) for the result listed as Test 1 is shown in Figure 3.
It is worth noting that R a ≥ 90 is generally considered as excellent colour performance, and LER ≥ 300 lm/W exceeds that of many other (efficient) light sources. Table 1 shows that significant step decrements in the weight e (the R min weighting) led to fairly small losses in the colour performance of the mixtures, but to noticeable gains in LER (and hence, efficiency in use). It can be seen that η rad ( Figure 5), and hence also f fit , undergoes significant swings during the early stages of the process (G < 300). It then converges to just below 340 lm/W at G = 336. In this instance, the population has converged around G = 400, and all parameters are only fine-tuned thereafter. These curves indicate that the process could, in fact, have been terminated at G = 500, with low likelihood of loss.

Optimisation of mathematical mixtures for LER and CRI (CIE R a )
It was decided to extend the above approach to an unconstrained investigation of SPD mixtures defined by the following mathematical functions: • Gaussian • rectangular
Influence of different f fit weights on the optimization of LED mixtures. 45 nm (full width at half maximum, or FWHM) [27]. Apart from these changes, the procedure was essentially the same as that used in Section 4.1, and the same fitness function was used. A total of 36 optimisations were performed, using 12 different combinations of f fit weightings for each of the three functions. The following extract from this set of results shows the SPDs obtained with the weightings: a = 1; b = c = d = e = 0. See Figure 6.
It was intended that the unconstrained approach will help to identify those wavelength combinations that will lead to optimum SPDs. The findings relating to the spectra in Figure 6 are summarised in Table 2.
This is too small a sample from which to generalise; however, it is possible to make the following observations. The colour performance of these three mixtures is exceptionally good (other than the high CCT for G1) possibly at the expense of relatively moderate LER values. It is probable that the Gaussian and triangular shapes are more realistic simulations of real physical spectra (whether originating from LEDs or phosphors); hence their corresponding centre wavelengths (λ 1 -λ 4 ) may be useful indicators for use in light-source design. See [27] for a more complete discussion.

Optimisation of LED mixtures for LER and CQS (NIST Q a )
This was again a constrained optimisation process. It was based on the same set of Luxeon™ LEDs as used in Section 4.1, and also used tetrachromatic mixtures. The optimisation process [23] was also broadly similar to that in Section 4.1, and it used a similar interface, but the software was rewritten to compute the performance of candidate SPDs in terms of CQS parameters, specifically the average colour quality index, Q a , and the minimum, Q min (the lowest individual index, Q i ) for every candidate SPD.
The following fitness function (Eq. 5) was employed: where a, b, c are the weights controlling the influence of Q a , LER and Q min on the optimisation of the LED mixtures. A selection of the results of the optimisation of the LED mixtures using various weightings is shown in Table 3. 98  93  96  228  80  12  7870  456  524  591  671   R1  97  90  94  317  72  12  3542  445  510  570  625   T1  98  95  97  303  84  12  4133  460  525  585  645   Table 2.
Results for the mixtures of mathematical functions shown in Figure 4.

Optimisation of LED mixtures for specific CCT values
The previously mentioned techniques (used in Section 4.1-4.3) did not take account of the CCTs of the spectra except as values to be calculated following the optimisations. Because of the importance of the CCT to lighting designers and users, it was recognised that this parameter needed to be incorporated into the optimisation procedure.
We therefore designed a new optimisation tool [29], again based on differential evolution, which puts CCT at the centre of the process and then proceeds to optimise colour rendering while maintaining a close tolerance to the target CCT value. We selected three CCT values to illustrate the effectiveness of the process.
The optimiser was presented with a specific set of four monochromatic LED spectra selected from the Luxeon™ range. The objective was to minimise the average colour difference for a set of colour samples which, in principle, could be any suitable set of colours. We chose to use the 14 test colours specified in CIE13.3 [13] since they constitute a well-known and widely used set.
The basis of the selection process in our algorithm was the colour difference of specific surface colours as they appear under the candidate spectrum and under the reference spectrum of the same CCT. In each new generation, the offspring solutions were evaluated on the basis of minimising the colour difference ΔE 00(Avg) calculated using the CIEDE2000 colour difference formula [31]. Hence, the algorithm was designed to search for a spectrum with the lowest colour differences. The optimum solution was determined after having performed G generations (typically 1000); that is, the best solution in generation G is accepted as the best white-light spectrum. Figure 7 shows the results in the form of the SPDs realised by the LED mixtures, and their performance is summarised in Table 4. Note that ΔE 00(Avg) represents the average of 14 colour differences (for the 14 colours in the CIE set) and Δ(u 0 ,v 0 ) is the chromaticity difference in CIE 1976 coordinates between the target CCT and that achieved by the LED mixture. The other listed parameters were computed after the completion of each optimisation run.   Table 4.
Optimizations of tetrachromatic LED mixtures with specified CCTs.
The above approach was extended to a set of optimisations in which Gaussianshaped spectra of either 25 or 50 nm bandwidth, but with unconstrained peak intensities and wavelengths, were optimised in the same way as the LED mixtures. Table 5 shows the best CRI performance for 4-band (25 nm) Gaussian mixtures, along with the other properties as listed in Table 4, and there is an evident degree of conformance between the two sets of results.

Optimisation of laser diode mixtures using TM-30-15 (IES R f )
Our investigations in this field [28,32] were inspired by the work of Neumann et al. [33] who successfully demonstrated a white-light 4-laser mixture with acceptable colour properties expressed in terms of the CRI and CQS metrics. Our earlier paper [28] also used CRI and CQS as measures of colour rendition. These approaches are now regarded as suspect (particularly in respect of mixtures of ultra-narrow bands such as lasers) on the grounds of incomplete sampling of wavelength and colour spaces.
Our purpose in [32] was to optimise similar mixtures of solid-state lasers, and to utilize the more up-to-date method of the IES colour fidelity index R f [15,16].
A new optimisation tool was designed to implement the complex sequence of non-linear calculation steps in the evaluation of colour fidelity for any given SPD. We then developed a new fitness function (Eq. 6) based on the IES parameters as well as the LER, to evaluate every candidate spectrum produced in the process.
f ¼ aR f þ bR f min þ cR f skin þ dη rad (6) where R f = average fidelity index for all 99 colour samples; R fmin = value of the lowest-scoring individual R f (single colour sample); R fskin = average index for two individual colours selected as representative of human skin; η rad = value of LER in lm/W; a, b, c, d are the respective weighting factors.
For the purposes of optimisation, each laser was simulated by a pseudo-delta function (1-nm bandwidth) at the centre wavelength of its output. Optimisations were performed in both constrained (using commercially available laser diode wavelengths) and unconstrained modes.
The best result (with four real laser wavelengths) was R f of 84 with LER 364 lm/W, which indicates the feasibility of the mixed-laser approach to provide highly efficient, energy-saving light sources. The unconstrained mixtures showed that the prospects will be further enhanced by potential future developments in semiconductor lasers, with the possibility of producing R f of 86 with LER of 380 lm/W. The detailed properties for these two mixtures are given in Table 6.
Compared with LEDs, lasers have the advantage of higher conversion efficiency (from electrical input to radiant output). This, combined with the very high prospective LER values, will make future laser mixtures exceptionally attractive in terms of energy conservation. The colour properties are not as good as for LEDs, but they are nevertheless regarded as sufficiently good for many types of lighting situations. Lasers are not yet considered to be practical as white-light sources but, with the potential for new developments in solid-state visible laser sources, their future prospects must be very strong.

Conclusions
We have here demonstrated the power of the differential evolution algorithm in the intelligent design of light source spectra, based on both LEDs and laser diodes. It provides a simple, flexible and effective solution in the elusive search for the balance of the LER and the colour rendition properties in optimised light sources. We recommend this technique to anyone engaged in the optimum design of lightsource spectra.
We feel confident that our method can also be readily adapted to other types of optimisation problem, wherever suitable elements for the fitness function can be readily identified.