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Abstract
The advancement of modern lighting technologies has led to many revolutions in lighting efficiency and presentation. The progression from filament bulbs, to CFL, and now LED technologies have produced a bounty of energy-efficient lighting options for design engineers and consumers. As the light-producing elements of a lighting fixture improve, the limiting factor in efficient illumination is no longer the light source, but the optical system itself. There are many characterization methods and standards for defining light for illumination in terms of color and human response. With concerns of how things like light pollution and energy requirements impact our society and the world around us, it is critical to understand how well a lighting fixture can illuminate a desired area while minimizing light lost to the environment and maximizing the total radiative intensity (radiance) of a space. This work presents two figures of merit, one for over-illumination and another for under-illumination, to characterize the optics of a lighting system based on a ray tracing methodology. Five common simplified optical design, with four varying beam angles, were simulated to present these new figures of merit. Results showed that common imaging optical systems such as parabolic and ellipse reflectors struggled to produce a well-lit area without over illumination, while nonimaging alternatives like the compound parabolic and compound elliptical reflectors were able to reach the thermodynamic ideal of a fully illuminated area without light lost to the environment. This work hopes to inform illumination engineers and lighting designers to help improve their optical design to maximize performance and minimize waste.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
There have been many techniques and standards developed over decades and centuries to characterize the quality of light. The SI units of measuring light such as the lumen and the candela themselves have been a metric to compare any light fixture to that of a simple candle. Over time more complex means to characterize lighting quality have been developed. Standards such as those set by the International Commission on Illumination (CIE). Many of these standards like the color rendering index CIE Ra value, corresponding color temperature, and the CIE 1931 color space [1] rely primarily on spectral distribution of light and how the human eye perceives color. Other human factor approaches such as brightness requirements for certain working environments [2–5] and studies on the spectral adaptation of human circadian rhythm [6–8] have seen significant study. Some work has been done to develop optical systems that produce uniform illumination across a target in terms of color [9–13] and total radiative intensity (radiance) in units of W m−2 sr−1 [14–21]. A significant amount of work has been done on the concept of Etendue, or conservation of four-dimensional phase space volume (X, Y, direction cosine of X, direction cosine of Y). The concept of etendue dilution [22] has been used define when the etendue between two surfaces is not fully utilized or fully filled for the purpose of energy transfer. Etendue dilution gets closer to the idea of what ideal illumination should be, but it can be non-intuitive and difficult to calculate. Industry standards such as beam lumens or field lumens are sometimes used for general illumination applications. Though useful and easy to understand, the beam and field lumens concepts don’t necessary define what is “ideal”, or the upper limit of what is achievable in a design.
Each of these characterizations of light are important, but many of them are subjective and based on the variability of the environments being illuminated, and the individual perception of a human viewer. Though beam lumens, field lumens, and etendue dilution come close to the answer, none of the other approaches seem to address major issues facing our society when it comes to illumination, like light pollution and energy waste [23–27]. A new approach is necessary to define the thermodynamic limits of illumination for any design as to minimize wasted light while maximizing the illumination of a defined space.
2. Method
2.1 Thermodynamics of illumination
To establish a figure of merit that categorizes ideal geometrical illumination, the concepts of under illumination and over illumination need to be quantified. An ideal illuminator should not miss any part of the desired illumination area, nor lose any light to space outside the desired illumination area (Fig. 1). These two concepts can be determined independently and then coupled together to determine the performance of a lighting system. This section is intended to outline the thermodynamics of both over illumination and under illumination based on the concepts of radiative transfer between surfaces.
Fig. 1. Diagram of (a) the concepts of ideal, over, and under illumination for a desired illumination area, and (b) an illumination setup with (1) the target illumination area (red), (2) the aperture of a lighting fixture (green), and (3) the light source plane (blue).
First, we examine the concept of over illumination by considering the simplified lighting problem outlined in Fig. 1(b). An efficient lighting system should ensure that none of the light misses the target. In other words, all the light emitted from the light emission plane 3 reaches the target illumination area 1. The total luminous power, ${Q_{31}}$, being emitted by 3 that reaches 1 can be defined as
where ${A_3}$ is the total emitting area of the 3 [units of m2], ${E_3}$ is the total emissive power of 3 [units of lumens/m2 or lux], and ${P_{31}}$ is the probability of radiation leaving 3 that reaches 1. ${P_{31}}$, though similar to view factors, ${F_{31}}$, used in radiative heat transfer, is different in that radiative transfer probabilities include any radiation that is either reflected, refracted, or transmitted through an optical system [28]. By examining Eq. (1) it becomes readily apparent that for all the radiation emitted from 3 to reach 1 with none of light missing the target, ${P_{31}} = 1$. Assuming that all the radiation emitted from 3 passes through the aperture at 2, ${P_{32}} = 1$, then if ${P_{31}} < 1$ only a fraction of the total radiation emitted from 3 reaches 1. This implies that 1 is either over illuminated or the light emitted by 3 is missing 1 entirely. Any lighting system illuminating a target area will have a radiative probability defining over illumination of the form, with ${P_{31}} = 1$ meaning no over illumination. Any value less than one in Eq. (2) signifies that light emitted from the source is missing the target entirely.Under illumination is qualified by the light flux emitted from a source not being distributed across the total target area. To do so in a way that maximizes the full illumination of the target area, the following thought experiment is proposed. Consider a light source with a fixed emissive power, ${E_s}$, placed in two configurations:
The maximum light flux at the target illumination area is achieved when the light source is placed at the aperture, as in configuration A. However, configuration A often loses a significant amount of light to the environment since ${P_{21}} \ne 1$. Applying the theory of reciprocity used in radiative heat transfer calculations (${A_2}{P_{21}} = {A_1}{P_{12}}$, and ${A_1}{P_{31}} = {A_1}{P_{13}}$) to configurations A and B, Eq. (3) and Eq. (4) respectively, we find that In order for configuration B to achieve the maximum light flux on the target illumination area, Eq. (5) and Eq. (6) must be equal to each other, requiring that ${P_{13}} = {P_{12}}$. To understand how ${P_{12}}$ and ${P_{13}}$ relate to under illumination, their physical meaning can provide some insight. ${P_{12}}$ and ${P_{13}}$ are the probability that radiation emitted from 1 will reach either 2 or 3 respectively, so ${P_{13}} < {P_{12}}$ signifies that some of the light from 1 that passes through 2 does not reach 3. This scenario can be thought of as rays emitted from the edge of the target illumination area that pass through the aperture and miss the light emission plane by either being reflected or refracted back out of the optical system (Fig. 2). By taking the ratio of ${P_{13}}$ to ${P_{12}}$ the amount of under illumination can be quantified as any value less than one (Eq. (7)). Both Eqs. (2) and (7) can be used together to quantify how well a lighting system illuminates a desired area, via over illumination and under illumination respectively. Though a blackbody assumption was used in the development of both Eqs. (2) and (7) the theory of radiative probabilities does not require the light source to be an ideal blackbody. If a source has a specific angular distribution of radiative intensity (sometimes called radiance), optics should be designed to fit such a source. The radiative transfer probabilities will still apply since they are based purely on geometry and material properties. Any lighting configuration with a defined size and location for the emission source, aperture, and target illumination area can be compared using both ${P_{31}}$ and ${P_{13}}/{P_{12}}$ by plotting them on a graph such as shown in Fig. 3. An ideal lighting system will have values for both equations equal to one. It is interesting that the ideal optics from a thermodynamic point of view will achieve both at the same time. One can easily design a device that satisfies Eq. (2) but not Eq. (7), or the other way around. Only the ideal optics can avoid both over illumination and under illumination.Fig. 2. Diagram of the physical significance of the radiative probabilities ${P_{12}}$ (red dashed line) and ${P_{13}}$ (blue dashed line) and how they relate to under illumination.
Fig. 3. ${P_{31}}$ vs ${P_{13}}/{P_{12}}$ showing the bounding limits of illumination with the ideal located at ${P_{31}} = 1$ and ${P_{13}}/{P_{12}} = 1$
2.2 Simulation and validation
The concepts of both ${P_{31}}$ and ${P_{13}}/{P_{12}}$ are ideally suited to validation through ray tracing. The definition of radiative transfer probability as defined in section 2.1 is the amount of light emitted by one surface that reaches another, through any optical system including reflected, refracted, and transmitted light. Any ray tracing technique used in most modern ray tracing software can easily determine these radiative transfer probabilities by emitting a known number of rays from one surface and counting the number of rays that impact the other surface. The ratio of these two numbers of rays gives the radiative transfer probability as shown in Eq. (8). Equation (8) deliberately discounts non-ideal reflectance and transmittance through an optical system. If one considers any optics problem from a ray tracing point of view, each ray in a raytracing simulation is a small volume of etendue in the four-dimensional phase space (X, Y, direction cosine X, direction cosine Y). By having less than 100% optical properties one effectively reduces this etendue/phase space volume connecting the source and the target. The goal of Eq. (8) is to define the upper bounds of what is theoretically possible for an illumination scenario. Any optical inefficiencies can be considered within a raytracing simulation to further define what a realistic performance might be.
Fig. 4. Diagrams of the five simple reflector profiles used in benchmark raytracing simulations testing the ${P_{31}}$ and ${P_{13}}/{P_{12}}$ figures of merit. (a) tapered/conical, (b) elliptical, (c) parabolic, (d) CPC, and (e) CEC reflectors. Blue depicts the light source area, green depicts the aperture area, gray depicts the reflector profile, red depicts the ray path, and the dotted arrow shows the beam angle for each design. Two-dimensional designs were used as shown in the figure, and three-dimensional designs were developed by revolving the profiles about a central axis.
3. Results
Results are compiled from three ray tracing simulations per each reflector profile and beam angle combinations. The three simulations per design were conducted to calculate the parameters in Eqs. (2) and (7). Each simulation varied the distance between the target plane and the aperture plane from 0 to 2 m in 0.025 m increments. The goal of varying the target distance was to validate the usefulness of using ${P_{31}}$ and ${P_{13}}/{P_{12}}$ as figures of merit for characterizing under and over illumination. All simulations where run in LightTools raytracing software using five million rays per simulation.
Figure 5 displays both ${P_{31}}$ and ${P_{13}}/{P_{12}}$ as a function of target distance from the aperture plane, and ${P_{31}}$ vs ${P_{13}}/{P_{12}}$ for each of the five reflector designs, with Fig. 5 a through e correlating to the designs designated in Figs. 4(a)–4(e) respectively. Each design beam half-angle is shown as a different color in Fig. 5 with blue, green, red, and purple representing 15 degrees, 30 degrees, 45 degrees, and 60 degrees. As a more direct comparison between each reflector design, Fig. 6 shows a plot of ${P_{31}}$ and ${P_{13}}/{P_{12}}$ values at the desired target distance of 1 m.
Fig. 5. Results of the raytracing simulations for the 2D reflector profiles calculating P31 and P13/P12 by varying the target distance L from the aperture. The first column shows P31 in dashed lines and P13/P12 in solid lines with the blue, green, red, and purple representing the 15, 30, 45, and 60 degree beam half-angles respectively. Figures 5(a)–5(e) correspond to the reflector designs depicted in Figs. 4(a)–4(e). The second column shows P31 vs P13/P12 for each design.
Fig. 6. Results of P31 vs P13/P12 for all (a) 2D and (b) 3D reflector designs. The points plotted are the values calculated from raytracing simulations for a target distance of 1 m from the aperture of each luminary. Colors distinguish the reflector profile and the symbols distinguish each beam half-angle.
4. Discussion
The most interesting trend in the simulation data was the noticeable transition seen between under illumination and over illumination. For an ideal design, this transition should occur at the specified target distance (Fig. 7). Except for the tapered and parabolic reflectors, many of the simplified 2D designs studied saw this transition occurring at or near the target distance of 1 m (Fig. 5). In the 2D designs presented in Fig. 5, where the luminary is faced directly at the target illumination plane, under illumination was found to be a minimum of the ratio of source area to aperture area, seen as the dashed line in Fig. 7. However, this was not the case for the 3D parabolic and elliptical designs. Other lighting scenarios may produce less illumination on a target area than the source to aperture area ratio as well. Generally, this limiting ratio makes sense when considering how ${P_{13}}/{P_{12}}$ is determined, with the amount of light that can reach the source from the target being limited by the amount of light that can pass through the aperture area.
Fig. 7. Trends seen in the P31 and P13/P12 data as a function of distance from the aperture to target illumination area for an ideal illumination lighting fixture.
The direct comparison of each design (Figs. 6(a) and 6(b)) is perhaps more significant in that it demonstrates the effectiveness of the studied luminaries at illuminating a target at a defined distance. Almost all the designs saw little to no over illumination (${P_{31}} = 1$), but only the elliptical, CPC, and CEC designs in the 2D case and only the nonimaging CPC and CEC designs in the 3D case approached the ideal of no under illumination (${P_{13}}/{P_{12}} = 1$). Ideal illumination with little to no loss of light or under illuminated area was only produced by the CEC reflector for all beam half-angles simulated in both the 2D and 3D designs. The CPC did perform ideally for the 60 and 45 degree beam half-angles, but smaller beam angles fell short of ${P_{13}}/{P_{12}} = 1$. This is primarily because a CEC reflectors shape has design criteria that include a target area at a defined distance, while a CPC defines a target infinitely far away. The most striking difference between the 2D and 3D results shown in Fig. 6 is the drastic shift in performance of the parabolic and elliptical designs. This is primarily due to the source shape being cylindrical instead of a disk like the tapered/conical, CPC, and CEC sources.
5. Conclusion
The theory and simulations presented here demonstrated the thermodynamics of illumination, and how to quantify what makes an ideal optical system. Two proposed figures of merit, one for over illumination and another for under illumination, and the means by which to calculate them via ray tracing were shown. Various luminary reflector designs were presented and compared with these figures of merit to categorize which designs approach or reach the ideal thermodynamic limit to illumination. Nonimaging optics was the only design shown to meet the thermodynamic ideal for both figures of merit. Other, perhaps more complex designs not tested here may approach the ideal of ${P_{31}} = 1$ and ${P_{13}}/{P_{12}} = 1$, but only nonimaging optics can reduce light loss and maximize light transfer using a simple luminary design.
Though not entirely representative of all possible real-world lighting and illumination designs, the simulations presented were used as a selective demonstration of how the figure of merit for under and over illumination can be quantified for a variety of lighting scenarios. The goal of the authors has been to provide a means by which lighting and illumination engineers can characterize their designs. Future work hopes to develop a methodology for experimental validation of the thermodynamic limits to ideal illumination. It is hoped that the theory of illumination presented here can help to inform and develop lighting optics which can approach the thermodynamic limit.
Disclosures
The authors declare no conflicts of interest.
References
1. D. L. DiLaura, K. W. Houser, and R. G. Mistrick, The Lighting Handbook Reference and Application (2011).
2. H. W. Bodmann, “Quality of Interior Lighting Based on Luminance,” Light. Res. Technol. 32(1 IEStrans), 22–40 (1967). [CrossRef]
3. A. R. Bean and R. I. Bell, “The CSP index: A practical measure of office lighting quality as perceived by the office worker,” Light. Res. Technol. 24(4), 215–225 (1992). [CrossRef]
4. I. Moreno, “Illumination uniformity assessment based on human vision,” Opt. Lett. 35(23), 4030 (2010). [CrossRef]
5. K. M. Rebec and M. K. Gunde, “High-performance lighting evaluated by photobiological parameters,” Appl. Opt. 53(23), 5147 (2014). [CrossRef]
6. Q. Dai, W. Cai, L. Hao, W. Shi, and Z. Wang, “Spectral optimisation and a novel lighting-design space based on circadian stimulus,” Light. Res. Technol. 50(8), 1198–1211 (2018). [CrossRef]
7. J. Hye Oh, S. Ji Yang, and Y. Rag Do, “Healthy, natural, efficient and tunable lighting: Four-package white LEDs for optimizing the circadian effect, color quality and vision performance,” Light: Sci. Appl. 3(2), e141 (2014). [CrossRef]
8. G. E. Fernandes, J. Bohar, and J. Xu, “Spectral-Temporal LED Lighting Modules for Reproducing Daily and Seasonal Solar Circadian Rhythmicities,” 2017 IEEE Int. Conf. Smart Comput. SMARTCOMP 2017 (1430007), 1–6 (2017).
9. Z. M. Zhu, X. H. Qu, G. X. Jia, and J. F. Ouyang, “Uniform illumination design by configuration of LED array and diffuse reflection surface for color vision application,” J. Disp. Technol. 7(2), 84–89 (2011). [CrossRef]
10. A. Teupner, K. Bergenek, R. Wirth, J. C. Miñano, and P. Benítez, “Optimization of optical systems for LED spot lights concerning the color uniformity,” Thirteen. Int. Conf. Solid State Light.9190(September 2014), 91900J (2014).
11. A. Teupner, K. Bergenek, R. Wirth, J. C. Miñano, and P. Benítez, “Optimization of a merit function for the visual perception of color uniformity in spot lights,” Color Res. Appl. 40(3), 287–296 (2015). [CrossRef]
12. A. Teupner, K. Bergenek, R. Wirth, J. C. Miñano, and P. Benítez, “Merit function for the evaluation of color uniformity in the far field of LED spot lights,” Proc. SPIE 9003, 900303 (2014). [CrossRef]
13. D. K. Son, E. B. Cho, I. Moon, Y. Park, and C. G. Lee, “Development of an illumination measurement device for color distribution based on a CIE 1931 XYZ sensor,” J. Opt. Soc. Korea 15(1), 44–51 (2011). [CrossRef]
14. Y. Ding, X. Liu, Z. Zheng, and P. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958 (2008). [CrossRef]
15. Z. Zhenrong, H. Xiang, and L. Xu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48(35), 6627 (2009). [CrossRef]
16. Z. Su, D. Xue, and Z. Ji, “Designing LED array for uniform illumination distribution by simulated annealing algorithm,” Opt. Express 20(S6), A843 (2012). [CrossRef]
17. R. Hu, Z. Gan, X. Luo, H. Zheng, and S. Liu, “Design of double freeform-surface lens for LED uniform illumination with minimum Fresnel losses,” Optik (Munich, Ger.) 124(19), 3895–3897 (2013). [CrossRef]
18. R. Hu, X. Luo, H. Zheng, Z. Qin, Z. Gan, B. Wu, and S. Liu, “Design of a novel freeform lens for LED uniform illumination and conformal phosphor coating,” Opt. Express 20(13), 13727 (2012). [CrossRef]
19. J. Muñoz, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46(3), 033001 (2007). [CrossRef]
20. J. M. Gordon, “Tailored edge-ray designs for uniform illumination of distant targets,” Opt. Eng. 34(6), 1726 (1995). [CrossRef]
21. J. M. Gordon, P. Kashin, and A. Rabl, “Nonimaging reflectors for efficient uniform illumination: errata,” Appl. Opt. 32(13), 2303 (1993). [CrossRef]
22. N. Shatz, J. Bortz, and R. Winston, “Thermodynamic efficiency of nonimaging concentrators,” Proc. SPIE 7423, 742308 (2009). [CrossRef]
23. F. Falchi, P. Cinzano, C. D. Elvidge, D. M. Keith, and A. Haim, “Limiting the impact of light pollution on human health, environment and stellar visibility,” J. Environ. Manage. 92(10), 2714–2722 (2011). [CrossRef]
24. F. Hölker, T. Moss, B. Griefahn, W. Kloas, C. C. Voigt, D. Henckel, A. Hänel, P. M. Kappeler, S. Völker, A. Schwope, S. Franke, D. Uhrlandt, J. Fischer, R. Klenke, C. Wolter, and K. Tockner, “The dark side of light: A transdisciplinary research agenda for light pollution policy,” Ecol. Soc. 15(4), art13 (2010). [CrossRef]
25. C. D. Galatanu, M. Husch, L. Canale, and D. Lucache, “Targeting the Light Pollution: A Study Case,” 2019 IEEE Int. Conf. Environ. Electr. Eng. 2019 IEEE Ind. Commer. Power Syst. Eur. (EEEIC / I&CPS Eur.)1–6 (2019).
26. T. Gallaway, R. N. Olsen, and D. M. Mitchell, “The economics of global light pollution,” Ecol. Econ. 69(3), 658–665 (2010). [CrossRef]
27. F. Falchi, R. Furgoni, T. A. Gallaway, N. A. Rybnikova, B. A. Portnov, K. Baugh, P. Cinzano, and C. D. Elvidge, “Light pollution in USA and Europe: The good, the bad and the ugly,” J. Environ. Manage. 248(February), 109227 (2019). [CrossRef]
28. B. Widyolar, L. Jiang, and R. Winston, “Thermodynamics and the segmented compound parabolic concentrator,” J. Photonics Energy 7(2), 028002 (2017). [CrossRef]
