One-time ray-tracing optimization method and its application to the design of an illuminator for a tube photo-bioreactor

This study details a one-time ray-tracing optimization method for the optimization of LED illumination systems [S.-C. Chu and H.-L. Yang, “One-time ray-tracing method for the optimization of illumination system,” in Proceedings of International Conference on Optics in Precision Engineering and Nanotechnology (icOPEN, 2013), 87692M]. This method optimizes the performance of illumination systems by modifying the light source’s radiant intensity distribution with a freeform lens, instead of modifying the illumination system structure. Because illumination system structures are unchanged in the design process, a designer can avoid the common problems faced when designing illumination systems, i.e., the repeated and time-consuming ray-tracing process when optimizing the illumination system parameters. The easy approaches of the proposed optimization method to sample the target illumination areas and to divide the light source radiant intensity distribution make the proposed method can be applied to both direct-lit and non-direct-lit illumination systems. To demonstrate the proposed method, this study designs an illuminator for a tube photo-bioreactor using the proposed one-time ray-tracing method. A comparison shows that in the designing of the photo-bioreactor, tracing all rays one time requires about 13 hours, while optimizing the light source’s radiant intensity distribution requires only about twenty minutes. The considerable reduction in the ray-tracing time shows that the proposed method is a fast and effective way to design illumination systems. ©2014 Optical Society of America OCIS codes: (220.2945) Illumination design; (220.4298) Nonimaging optics. References and links 1. X. Hu and K. 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Introduction
Because of the advantages over incandescent light sources, such as lower energy consumption, longer lifetime, smaller size and faster switching, light-emitting diodes (LEDs) are now widely used in road lighting [1,2], automotive lighting [3,4], interior lighting [5] and LCD backlighting [6].LED applications often achieve the specified illumination at the target plane.For example, researchers have used LED array lighting (LAL) to achieve uniform illumination [7][8][9][10].Among these LED uniform illumination studies, the "reversing design method" that Liu et al. proposed for slim direct-lit LED backlighting [7] has attracted attention for its potential to be modified as a fast design approach for all LED-based illumination systems.In Liu's method, the LED light intensity distribution curve (LIDC) is described in the form of a θ-polynomial, where θ is the emitting angle between the ray and the LED chip normal.At first, this method formalizes the illuminance of an arbitrary point on the target illumination plane with the LED LIDC.After that, it achieves illuminance uniformity of the central target region by Sparrow's criterion, and control of the side point illuminance by controlling the illuminance ratio between the side point and the central point on the target plane.In this method, more side points (verification points) are added during the iteration process until the system performance meets the specified requirements.The key difference between Liu's method and other methods is that Liu's method achieves LED array illuminance uniformity by modifying the LIDC of the LED array, and not by modifying the configuration of the LED array [7].The development of the freeform lens design and fabrication technology now allows LEDs to be easily integrated with freeform lenses to accomplish high-efficiency illumination control [11][12][13][14][15]. "To optimize the illumination system performance by modifying the light source's illuminating property" is a concept of great possibilities in the design of LED-based illumination systems.
In many LED-based illumination systems, the exiting light from the LED interacts with other optical elements before arriving at the target illumination plane.For example, in edge-lit LED backlighting, LEDs are usually combined with a light guide plate (LGP) to achieve high illuminance uniformity [16][17][18].The illumination performance of such non-direct-lit systems is not easily directly formalized by light source's illuminating property as is the direct-lit lighting system, as in Liu et al. [7]; thus, Liu's method is not workable for these non-direct-lit LED lighting systems.To describe the performance of such non-direct-lit LED lighting systems, an optical designer must usually do Monte Carlo ray-tracing, which requires tracing one hundred thousand rays or more for a one-time ray-tracing simulation.During the designing process, each time a designer modifies the structure parameters of an illumination system, he or she must do the ray-tracing again to confirm the illumination system property.Increased computer calculation speed has helped save design time.Also, commercial Monte Carlo ray-tracing software has ray-tracing techniques which reach the same level of accuracy in the shortest ray-tracing time, e.g., the "backward ray tracing" method of the software LightTools [19] and the "importance sampling" method of the software TracePro [20].However, the more complex an illumination system, the more rays need to be traced in the simulations, and it becomes much harder to sustain the repeatedly time-consuming ray-tracing process in an illumination system's design.
This paper details an approach for designing LED-based illumination systems, the onetime ray-tracing optimization method [21].The key concept of this method is the same as Liu's "reversing design method" [7], i.e. this method optimizes illumination system performance by modifying the light source's illuminating property with a freeform lens, instead of modifying the illumination system structure.The dividing/sampling approach of the target illumination area and the light source of the proposed method makes it workable for non-direct-lit LED illumination systems, as well as direct-lit LED illumination systems.In the design process of the proposed method, the system structure remains unchanged as the designer does not need to trace the rays more than once.By means of this one-time raytracing and a general optimization process, a good solution for an LED-based illumination system design can be quickly found.In this study, the proposed method was used to design an illuminator for a tube photo-bioreactor which would provide uniform illuminance at two target illumination surfaces inside the tube photo-bioreactor [22].It should be noted that the application of the proposed method is not restricted to uniform illumination, but can also be applied in the design of other illumination systems with specified illumination requirements simply by modifying the merit function in the optimization process.
The paper is organized as follows: in Section 2, the one-time ray-tracing optimization method is detailed; in Section 3, the proposed method is applied to the design of a tube photobioreactor for cultivating microalgae [23,24]; in Section 4, some concerns regarding the use of the one-time ray-tracing optimization method are discussed; and, finally, in Section 5, a brief summary of this study is given.

One-time ray-tracing optimization method
This section details the process of using the "one-time ray-tracing optimization method."When a designer has an initial design for an illumination system, he or she optimizes the system performance by modifying the light source's illuminating property, rather than modifying the illumination system structure.In this study, the light intensity distribution curve (LIDC) of the light source is a variable of the system optimization.In this section, the design of an edge-lit LED lighting system has been used as an illustration to explain the process of the proposed method.Figure 1 shows a schematic diagram of a common edge-lit LED illuminator.In order to easily explain the method, a specific design goal of an illuminator "maximizing the illuminance uniformity of a target illumination area" was set.

Finding effects of all variables separately
In this method, the system performance is optimized by modifying the LIDC of the light source; that is, the proposed optimization method selects the LIDC as the variable, and then describes the illumination system performance with a merit function related to the LIDC to be optimized.Referring to Fig. 2, the radiant intensity distribution of the light source is divided into M source zones (S-zones), S i , by its light-emitting angle θ to a reference direction denoted by unit vector n .This study lets n be along the surface normal of the LED chip, where each S-zone indicates an emitting angle interval of the light source.In the one-time ray-tracing process, all light source zones are set with a unit radiant intensity, as the purpose of the one-time ray-tracing is to simply find the characteristics of the initial illuminator design.Note that, in order to easily explain the method, in this study the LIDC has azimuthal symmetry; however, depending on the requirement of each illumination design, the LIDC can also be out of azimuthal symmetry.LED applications often achieve the specified illumination at the target plane [1][2][3][4][5][6][7][8][9][10][11].In this sample design, the illuminator hopes to achieve a uniform illuminance distribution at a target illumination area, as Fig. 1 shows.This method divides the target illumination area into N target zones (T-zones), T j .
Specifically, this method separately traces rays from each source zone, rather than tracing rays from all light source zones simultaneously.The corresponding average illuminance at each target zone resulting from each light source zone is also separately recorded.The average illuminance at the j th target zone T j resulting from the i th light source zone S i is denoted by F j i , where the subscript indicates the T-zone number and the superscript indicates the S-zone number (i.e., emitting angle interval).The resulting effect (here, average illuminance) at each T-zone resulting from each S-zone is described by the M × N characteristic matrix F: where the characteristic matrix F contains the property of the initial design of the illumination system structure.

Optimizing system property
For optimizing the system property by modifying the LIDC of the light source, this method introduces a flux weighting array G: where component g i describes the weight of the total-emitting-flux from each light source zone, S i .The value, F j i , which is the average illuminance at the j th target zone T j resulting from the i th light source zone S i , is proportional to the total-emitting-flux from light source zone S i .Thus, it is easy to find the resulting average illuminance distribution at all target zones from a flux-weighted light source.As Eq. (3) shows, the flux weighting array G multiplied by the characteristic matrix F gives a weighted illuminance array P: ... , where component gives the prediction of the average illuminance at each target zone using a modified light source with a weighted flux.That is, as one adjusts the flux weight of each light source zone S i , one can easily find the average illuminance at the target illumination area without additional ray tracing.The weighted flux of the light source will be achieved by a modified LIDC later.The next step is finding a suitable flux weighting array G to optimize the illumination system.The process is addressed as follows.First, this method describes the illumination system performance by a meaningful merit function, which is expressed with the flux weighting array component g i .The merit function should be specified by the designer according to the requirements of the illumination system.Second, simply by using common optimization algorithms [25], the suitable g i can be found to minimize the merit function, i.e., to optimize the illumination system property.
For example, the illuminance uniformity at the target illumination area of the edge-lit LED illuminator is evaluated by the minimum-to-maximum average illuminance ratio R of all target zones (T-zones) at the target illumination area.That is, the ratio of the minimum average illuminance P min to the maximum average illuminance P max at the target illumination area is: where the R value ranges between 0 and 1, and the illuminance uniformity of the target illumination area increases as the R value increases.Substituting Eq. (3) into Eq.( 4) gives the R value in the form of a polynomial of multiple variables g i , i.e., R is the function of g i .This study uses the steepest descent method [26] to optimize the illuminance uniformity of the target illumination area by finding a proper flux weighting array G that maximizes the R value.Equation (5) shows the method by which the weighting array G is found: where γ is the step size of the steepest descent method, with the suffix of G denoting the number of iteration times.Figure 3 gives a simple illustration of operations of Eq. ( 5) in the steepest descent method.Note that the optimization goal of this design is to maximize the R value, so the change of the flux weighting array G here is along the positive gradient direction  ( ) The optimization process can find a suitable flux weighting array G M , which lets the target illumination area possess high illuminance uniformity.The weighting array G M corresponds to a specific modification of the total-emitting-flux distribution of light source zones S i : ( ) where the symbol Φ i (i = 1~M) denotes the original total-emitting-flux from each light source zone, S i .It is easy to find an LIDC which can produce the specific total emitting flux distribution, Φ(θ).Equation (7) shows the mathematical representation of the LIDC adopted in this method: ( ) where, for simplicity, the radiant intensity of each light source zone I i is a constant value.The I i value of each light source zone can be easily found using the energy conservation law: In this step, using a light source with a specific LIDC I(θ) in the initial design of an illumination system allows the target illumination area to possess high illuminance uniformity.The next section shows how to find a freeform lens to achieve the desired LIDC, I(θ).

Finding a freeform lens for the desired LIDC
The design and fabrication of the freeform lens has been well developed in recent years [12][13][14][15].Several freeform lens design methods can be modified to make LED possessing a specified LIDC, I(θ).This study adopted and modified Zhao's method, originally used for designing searchlights [27].Figure 4 shows the parameters used in designing the freeform lens.In this study, the left side of the lens surface is spherical and the right side is a freeform surface.In the freeform lens design process in this study, it was assumed the light source to  Referring to Fig. 4(a), Snell's law gives the relation between the incident ray and the refractive ray in the vector form as: where n is the refractive index of the material of the freeform lens; and vectors N  , O  and I  are the unit direction vectors, which denote the normal vector at ray-incident point P, the incident ray and the output ray, respectively.The emitting angle and adjusting angle of a ray before and after passing through the freeform lens are denoted by the symbols φ and θ, respectively.The LIDC of the original Lambertian point source follows the cosine relationship I(φ) = I 0 cos(φ), where the symbol I 0 denotes I(φ = 0).The emitting angle φ of the original Lambertian point source ranges between φ = 0° and φ = 90°.Referring to Fig. 4(b), the right side of the freeform surface design starts from the initial point P 0 situated at (z,y) = (d,0).At this point, the initial ray emitting angle φ 0 and initial ray adjusting angle θ 0 are both zero.Equation ( 9), Snell's law, finds the freeform surface's normal vector of point P 0 , 0 N  .The next discrete point on freeform surface P 1 is located at the intersection point of 0 N  and the next incident ray 1 I  that possesses emitting angle φ 1 .Then, using the conservation of total flux before and after passing through the region of the freeform surface between intersection points P 0 and P 1 , we can calculate output ray vector 1 O  at point P 1 according to the specified adjusting intensity profile I(θ) given by Eq. ( 7).That is: where the suffix i refers to the suffix of the calculating discrete point P i .Equation (10) indicates that the total flux of emitting rays between emitting angles φ = 0 and φ = φ i is equal to the total flux of output rays between adjusting angles θ = 0 and θ = θ i .Now knowing the output ray vector 1 O  , Eq. ( 9) can be used to find the freeform surface normal of point P 1 .By repeating the above steps (i.e., finding i N  , P i and i O  ), all the discrete points of the freeform surface on the cross-section y-z plane can be found.The first freeform curve was constructed by directly connecting all discrete points sequentially.The normal of a line segment will be surface's normal vector of point P i , i.e., i N  .After that, rotating the freeform curve builds up the right side's freeform surface of the freeform lens.The constructed freeform lens can convert the Lambertian LIDC into a LIDC closed to the specified LIDC, I(θ), as specified in Eq. (7).Note that the resulting LIDC of the freeform lens-constructed light source is related to how many discrete points P i are used in building the right-side freeform surface.This study uses 345 discrete points to construct the first freeform curve, and the correlation coefficient between the resulting LIDC of the freeform lens-constructed light source and the specified LIDC is 96.6%.Note that the choosing of the initial z-position d of initial point P 0 will not influence the resulting LIDC of a freeform lens-constructed light source, since the design of the freeform lens using the perfect point light source as the light source.This study lets the initial d value be a unit value, 1.And then resizing the freeform lens according to the size of tube photobioreactor.

Summary of the one-time ray-tracing optimization method process
Figure 5 illustrates the process of using the proposed one-time ray-tracing optimization method to design a simple illuminator providing specified illuminance (here, uniform illuminance) at the target illumination area.The process has five steps.First, create the initial illumination system model using ray-tracing simulation software.Second, set the divisions/sampling of both the light source's radiant intensity and the target illumination area.Third, trace the rays of each source zone and record the flux of the each target zone separately.Fourth, describe the property of the target illumination area with a suitable merit function; then use the common mathematical optimization method to find a suitable light source weighting array for minimizing/maximizing the merit function accordingly.Finally, replace the light source of the initial design by a freeform lens-constructed light source, which possesses the specified LIDC according to the optimized flux-weighting array.Note that, when designing illumination systems for various purposes, three factors, the source divisions, target area divisions and merit function, may be changed according to the requirements of the illumination system's design.As a demonstration of the proposed method, in Section 3, the use of the proposed method to design the illuminator of a tube photo-bioreactor for microalgae cultivation is described.

Design of the illuminator for a tube photo-bioreactor using the one-time ray-tracing optimization method
We applied the one-time ray-tracing optimization method to design the illuminator for a tube photo-bioreactor system as a demonstration of the proposed optimization method.A common tube photo-bioreactor is composed of a gas supply, light source, stirring system and a glass tube for the fluid to flow through [22][23][24].It is usually used as a microalgal culture device which creates suitable conditions for growing microalgae.One of the most important issues considered in designing a tube photo-bioreactor is the illuminance inside the photo-bioreactor, which directly influences the growing efficiency and so the concentration of microalgae cells.Inhomogeneous illuminance (i.e., inhomogeneous microalgae cell concentration) will lead to light-absorption and light-scattering by the cells, which results in the decreased efficiency of the photo-bioreactor.Thus, the key goal in designing the tube photo-bioreactor illuminator is to achieve homogeneous illumination inside the photo-bioreactors.
Figure 6 shows the photo-bioreactor illuminator designed by Clemens Posten [23,24], where the illuminator is inside a glass tube containing fluid with microalgae.The illuminator is formed by an optical fiber ring light and a tube lightguide with rough side surfaces.The optical fiber-constructed ring light is guided into the tube acrylic lightguide, and then the rough surfaces of the tube lightguide diffuse the light into the photo-bioreactor.In this design, the illuminance uniformity inside the bioreactor depends on the design of the lightguide's surface roughness.Each time the surface roughness is modified, the designer must perform the time-consuming ray-tracing to verify the system's illumination property.Furthermore, complex tube lightguide surface roughness may make the mass production difficult.In the design of this study, the lightguide's surface roughness had simple and fixed micro structures, and the one-time ray-tracing optimization method was used to achieve homogeneous illumination inside the photo-bioreactor.

Initial design of the illuminator for a tube photo-bioreactor
Figure 7(a) depicts our design of the illuminator for a tube photo-bioreactor, comprised of a tube lightguide, several point-like sources and two tube reflectors situated at each end of the tube lightguide.The blue rectangle in Fig. 7(a) indicates the radial cross-section locations of Fig. 7(b), which shows the circular micro grooves on both tube lightguide surfaces.The tube lightguide contains rotational symmetry about the cylinder axis.The goal of this design was to achieve homogeneous illumination inside the photo-bioreactor; thus, in this study, two cylindrical target surfaces, the inner target surface and the outer target surface, as shown in Fig. 7(c), were set.As a demonstration of the proposed method, the goal of the tube illuminator design was to achieve homogeneous illuminance at these two target surfaces.More target illumination surface could be added in the design process to control the illumination in the entire photo-bioreactor.The parameters of this design were as follows.The material of the freeform lens is PMMA.The material of the tube lightguide was PMMA, with an inner diameter of R in = 40mm, outer diameter R out = 50mm, thickness t = 10mm and length L = 160mm.The lightguide's circular groove width was D = 0.1mm, with a center distance between two neighbor grooves of 2D = 0.2mm.The inner surface property of the tube reflector, length l = 7mm was set as a mirror surface possessing reflectivity of 98%.In the initial design, both sides of the tube reflector contained 48 circular Lambertian surface light sources of a small radius, 0.1mm, all situated 6 mm from the lightguide end.The length of the two cylindrical target surfaces was the same as the length of the tube lightguide L, and the distance from the two cylindrical target surfaces to the tube lightguide was d = 10mm.The surrounding medium of the tube illuminator was water, and the surrounding medium of all light sources in the reflector was air.In this study, the wavelength of all light sources was set as 550 nm.
Figures 8(a) and 8(b), respectively, show the unfolded illuminance distributions of the inner and outer cylindrical target surfaces of the initial design of the illuminator.The lateral axis values show the azimuthal angle of the cylindrical target surface corresponding to the tube axis, while the vertical axis values correspond to the z-position values of the two target surfaces.This study used the commercial software, LightTools [19], which adopts the Monte Carlo light-tracing method, to estimate the performance of the illuminator.Figure 8 shows the initial inhomogeneous illuminance distributions of the two cylindrical target surfaces from the initial illuminator design prior to the one-time ray-tracing optimization.In the design of the tube illuminator of the photo-bioreactor, this study used the minimum-to-maximum illuminance ratio R of the two target surfaces to evaluate system performance.When conventional point-like Lambertian surface sources were used as the light source of the illuminator, the R value of the two target surfaces of the initial design was 45.9% and 47.2%, respectively.The CV (coefficient of variation) values of the inner and the outer target surfaces were 0.3109 and 0.2520, respectively.In this study, the one-time ray-tracing optimization method was used to increase the illuminance uniformities of the two target surfaces of the tube photo-bioreactor illuminator.

Ray-tracing results
Before performing the one-time ray tracing method, the designer must make the divisions on both the light source emitting angle and the target illumination areas.Referring to Fig. 2, in this design the LIDC was restrictedly ranged from 3° to 39°, which ensured that all emitting rays from the later freeform lens-constructed light source could be guided by the tube lightguide with TIR (total internal reflection).Because the range of the light source emitting angle of this design is 36°, this study lets the division number of the light source zone be a multiple of 9 for simplifications.Increasing the number of the light-source division can help the later optimization of system performance.However, it will also directly increase the optimization time.In this design, the LIDC was divided into 18 equal zones.Later results will show that using 18 light source zones provides both acceptable optimization time and system performance in this design.
Referring to Fig. 7, both inner and outer cylindrical target surfaces were divided into 33 equal zones along the ray-guiding direction (i.e., z direction).Note that the choosing of the target area divisions should be able to reflect the illumination performance of the target area.Because the design goal here is to achieve homogeneous illumination at two cylindrical target surfaces, this design lets both cylindrical target surfaces be divided into equal zones.The division numbers of both cylindrical target surfaces are chose to be odd numbers for including the center illumination properties of two target surfaces.Similarly; increasing the division number of the target area can help the later optimization of system performance.However, it will also directly increase the optimization time.
Figures 9(a) and 9(b) show the cross-section average illuminance distributions resulting from each light source zone on the inner and outer target surfaces, respectively.For the onetime ray tracing, the light sources were set as point-like circular surface emitters with a 0.1mm radius.The illuminance contributions of each light source zone are described by different curves in Fig. 9.In the one-time ray-tracing process, all light source zones were set with the same unit radiant intensity, 1 W/sr, in order to determine the characteristics of the initial illuminator design.As Fig. 9 shows, the location of the main illuminance contribution from each light source zone was different.Thus, a suitable adjustment of the relative weight of each light source zone gave the two target surfaces homogeneous illuminance distributions.

Find suitable LED LIDC to optimize system performance
As discussed in the optimization process in Section 2.2, this design first introduces the light source flux weighting array G as the variable, and then represents the illuminance uniformity of the two target surfaces by the R value, i.e., the ratio of the minimum illuminance to the maximum illuminance of the target surface.The merit function of this design is defined as: where R in and R out denote the R value of the inner and the outer target surface, respectively.Simply running the steepest descent method to maximize the merit function quickly finds a suitable flux weighting array G M to give the two target surfaces high illuminance uniformity.Figure 10(a) shows the optimized LIDC I(θ), corresponding to the optimized flux weighting array G M .Figures 10(b) and 10(c) plot the resulting illuminance distributions of the two target surfaces when the tube illuminator uses point-like circular surface emitters, with a 0.1mm radius, possessing an optimized LIDC I(θ).In this situation, R in = 94.9% and R out = 95.3%, and the corresponding CV value of the inner and the outer target surface is 0.0125 and 0.0115, respectively.The results show that by using the one-time optimization approach one can easily find the suitable LED LIDC to optimize illumination system performance.

Final system properties with the freeform lens-constructed light source
Having found the optimized LED LIDC I(θ) for the tube illuminator design, the next step was to construct a freeform lens-constructed light source possessing light-emitting characteristics close to the optimized LED LIDC I(θ).The freeform lens-constructed light source was composed of a freeform lens and the common Lambertian LED chip.Figure 11(a) plots the cross section of the freeform lens constructed by the lens-building approach detailed in Section 2.3.In this design, the radius of the spherical side of the lens was set as 1 mm.This freeform lens-building approach was based on the assumption that the light source was a perfect point light source; however, because there is no perfect point light source in the real world, the LIDC of a real freeform lens-constructed light source will always depart from the ideal optimized LIDC I(θ) to some degree.The color lines of Fig. 11(b) show the normalized LIDC of the freeform lens-constructed light sources when using circular Lambertian LED chips of different sizes.The dotted line of Fig. 11(b) denotes the ideal situation, i.e., the optimized LIDC I(θ).The peak value of all LIDCs were normalized to 1 W/sr, where the symbol r denotes the ratio of the LED chip diameter to the diameter of the spherical side of the lens.Figure 11(b) shows that when the LED chip size was equal to or smaller than one-tenth of the diameter of the spherical side of the lens, the LIDC of the freeform lensconstructed light source deviated from the optimized LIDC by only a small degree.Figures 11(c) and 11(d) show the resulting illuminance distributions of the two target surfaces when using a freeform lens-constructed light source as r = 0.05, i.e., the diameter of the LED chip was 0.1 mm.In this situation, the minimum-to-maximum illuminance ratio of the target surfaces was R in = 85.3% and R out = 81.5%, and the corresponding CV value of the inner and outer target surface was 0.0763 and 0.1007, respectively.Comparing Figs.8(a) and 8(b) with Figs.11(c) and 11(d), the four figures show that the one-time ray-tracing optimization method could quickly find a good solution for an illumination system design.The R value of the inner and outer target surfaces of the tube illuminator substantially increased by 39.4% and 34.3%, respectively.The CV value of the inner and outer target surfaces of the tube illuminator substantially decreased by 0.2346 and 0.1513, respectively.However, comparing Figs.10(b) and 10(c) with Figs.11(c) and 11(d) shows that there was an obvious decline in the performance of the tube illuminator when the "point-like light sources possessing optimized LIDC" were replaced by "freeform lens-constructed light sources."In Section 4.1, we discuss why this happened and how to avoid it by using the one-time raytracing optimization method in designing illumination systems.For comparing our design with previous Posten's illuminator, we further plot Fig. 11(e), i.e., the cross-section illuminance distributions in the photo-bioreactor in water from the axis of the tube illuminator in six different distances: 10mm, 20mm, 30mm, 60mm, 70mm and 80mm.The R values of these six curves range from 0.68 to 0.94.Even though this study optimizes illuminance distributions at only two specified cylindrical target surfaces, this design also increases uniformity of illuminance distribution of the whole photo-bioreactor to some degree.The graph of the previous Posten's illuminator that similar to Fig. 11(e) could be found in Ref. 24, and the maximum R value of the graph of Posten's illuminator is around only 0.3.The comparison between two designs shows that the one-time ray-tracing optimization method could find a good solution to the illumination system design.The degraded performance of the tube illuminator did not actually result from the mis-design of the freeform lens.As Fig. 11(b) shows, when the Lambertian LED chip was smaller than one-tenth of the diameter of the spherical side of the freeform lens, the LIDC of the freeform lens-constructed light source was close to the optimized LIDC.More simulation results showed that a further decrease in the LED chip size of the freeform lens-constructed light source (e.g., r = 0.01) did not improve the performance of the tube illuminator.The degraded performance of the tube illuminator actually resulted from the additional interactions between the freeform lens and the reflected rays from the two tube reflectors.The physical size of the freeform lens-constructed light source was around 2.6 mm x 2.6 mm x 3mm, while the setting of the point-like light source in the one-time ray-tracing was a small, circular surface source 0.1mm in radius.The unexpected interactions between the freeform lens and the reflected rays from the two tube reflectors made the illuminance distributions on the two target surfaces resulting from each light source zone depart from those expected for the one-time ray-tracing (i.e., characteristic matrix and Fig. 9) to some degree.Further simulations show that the peak position and the min-to-max ratio of all curves in the Fig. 9 (i.e., cross-section average illuminance distributions resulting from each light source zone on the two cylindrical target surfaces) are changed while using the freeform lens-constructed light sources.These unexpected interactions were also the reason for the R-value difference between Figs. 10(b) and 10(c) and Figs.11(c) and 11(d).In the design example, the tube illuminator was a nondirect-lit LGP illumination system; thus, there existed the possibility of the second-time interactions between the freeform lens and the reflected rays.This issue should be minor in direct-lit designs and other non-direct-lit illumination designs.

Discussion
Reducing the size of the freeform lens-constructed light source is a way to increase the performance of tube illuminator design.Figure 12 plots the unfolded illuminance distributions of the two target surfaces of the tube illuminator when using a 10-time-contraction freeform lens-constructed light source (i.e., r = 0.05 and freeform lens of dimension 0.26 mm x 0.26 mm x 0.3mm).The R value of the two target surfaces of the tube illuminator was successfully increased to 95.3% and 95.1%, respectively, and the corresponding CV value of the inner and outer target surface was decreased to 0.0116 and 0.0170, respectively.Besides, further simulations also shows that the 10-time-contraction freeform lens-constructed light source got less change in the characteristic matrix than the original freeform lens-constructed light source.However, depending on the physical size of the illumination system, reducing the freeform lens-constructed light source may lead to a freeform lens size or an LED chip size of an unreasonably small value.A better and more reasonable solution would be to include the concept of "decreasing/preventing the possibility of the second-time interactions between the freeform lens with the reflected rays" in the initial design of a non-direct-lit illumination system when using the one-time ray-tracing optimization method.Taking the tube illuminator as an example, the initial design could let each point-like light source be sheltered by a cylindrical reflector of a physical size around that of an estimated freeform lens.This approach could partially decrease the possibility of the second-time interactions between the freeform lens with the reflected rays from the bottom of two tube reflectors, and thus decrease the system's performance difference between "using point-like light sources possessing optimized LIDC" and "using freeform lens-constructed light sources."

Advantages of using one-time ray-tracing optimization method to design illumination system
The main advantage of using the one-time ray-tracing optimization method to design illumination systems is that this design approach saves considerable time.Taking the tube illuminator design as an example, tracing all rays (i.e., nineteen million rays) has a one-time requirement of around 16 hours, while using the steepest descent method for finding the optimized LED LIDC takes about 20 minutes.It is obvious that repeated ray-tracing requires significantly more time than that needed to perform the steepest descent method optimization.The proposed one-time ray-tracing optimization method (Fig. 5) requires only the one time ray tracing in the design process, as the name suggests.Depending on the complexity of the illumination system, the ray-tracing time can be much longer than 16 hours.The one-time ray-tracing optimization method modifies the LED LIDC rather than modifying the optical structure parameters.Since the optical structures do not change in the design process, ray tracing does not need to be performed repeatedly.The proposed one-time ray-tracing optimization method provides a straightforward and fast approach for finding workable illumination system designs.As a reference to the reader, the specifications of the computer system used in this study were CPU (Central Processing Unit), Core i5; RAM (Random Access Memory), 4GB; and OS (Operating System), Windows 7, 64bit.The ray tracing was performed by the commercial Monte Carlo light-tracing software, LightTools [20], and the steepest descent optimization was performed by the software Matlab [28].It should be noted that increasing the division number of the light source and target area do help in the optimization of the system performance.However, it will also increase the optimization time simultaneously.Besides, each time when a designer changes light source division and/or target area divisions, he/she has to do ray-tracing again for finding the characteristic matrix of this optimization method.And to re-do ray-tracing will increase the amount of time for designing an illumination system.Since the ray-tracing time is much longer than the time to optimize the LED LIDC, we suggest designers to use large but reasonable initial divisions of light source and target area under an acceptable condition of his/her calculating equipment.
Another advantage of the one-time ray-tracing optimization method is that the massproduction complexity of the illumination system this method designed could be reduced.Let's take the tube illuminator design as an example and compare the two tube illuminator designs (Fig. 6 and Fig. 7).In Posten's illuminator design (Fig. 6), the illumination homogeneity of the illuminating area of the photo-bioreactor was achieved by modifying the surface roughness of the cylinder lightguide.The increased complexity of the lightguide's surface roughness may make the cylinder of the lightguide difficult to mass produce.The illuminator designed by the proposed method (Fig. 7) has the lightguide surface roughness as a simple micro structure for easy mass production.The illumination homogeneity of the illuminating area of the photo-bioreactor was achieved by optimizing the LED LIDC with the freeform lens-constructed light source.The well-developed freeform lens design methods and manufacturing techniques ensure the practicability of designing illumination systems with the one-time ray-tracing optimization method.

Summary
This paper detailed the "one-time ray-tracing optimization method" for the optimization of illumination systems.The key concept of this method is that of optimizing an illumination system by modifying the light source's illuminating property, rather than modifying the illumination system structure.Because the system structure is fixed, a designer need not perform the time-consuming ray tracing again and again in order to verify the illumination system performance.The proposed method avoids one of the common problems in designing illumination systems, i.e., the repeated and time-consuming ray-tracing process when optimizing the illumination system parameters.The easy approaches of the proposed optimization method to sample the target illumination areas and to divide the light source radiant intensity distribution make the proposed method can be applied to both direct-lit and non-direct-lit LED illumination systems.As a demonstration of this method for illumination system designs, in this study, we designed an illuminator for a tube photo-bioreactor (i.e., a non-direct-lit illumination system) using the proposed method.As to the design of the photobioreactor, tracing all rays one-time required around 13 hours, while optimizing the LED LIDC required only about 20 minutes.It was evident that the proposed method reduced the ray-tracing time and that this optimization method will speed up the designing of illumination systems.It should be noted that an improper initial design will definitely lead to the restriction of using the one-time ray-tracing optimization method in designing LED illumination systems.The initial design should at least be capable to illuminate every point on the target illumination area, such that the proposed optimization method can modifies the LED illuminating property to achieve the requirement of the illumination system.
Besides, with a suitable setting of the merit function of the optimization process, the method can achieve specific illumination design goals, as well as homogeneous illumination.This study used an azimuthal-symmetric freeform lens in the design of a tube illuminator possessing axial symmetry.When designing illumination systems without symmetry, designers can extend the degree-of-freedom in the freeform-lens shape (i.e., with arbitrary divisions in the light source's emitting angle), which would make the proposed method more efficient in designing illumination systems without symmetry.The details of the design of the asymmetric illumination systems by the proposed method will be presented in the near future.
Furthermore, this study only considers illumination system using light source of single wavelength.However; theoretically speaking, the proposed method can also be applied to design illumination system of broader spectra.Basically, it depends on the purpose of the illumination system and the development of the freeform lens-design technology.For example, in the design of tube illuminator, the resulting LIDC of the freeform lensconstructed light source is not sensitive to the light wavelength.(e.g., The correlation coefficient between two LIDCs of wavelengths: 550nm and 700nm, is 99.7%.)That is, using LED chip of broader spectra (e.g.550nm~700nm) in this design can also achieve uniform illuminance distributions at two cylindrical target surfaces.On the contrary; to design an LED illumination system for achieving a wavelength-dependent illumination by the proposed method still needs further investigating.It is possible to find an optimized LIDC of broader spectra (e.g.be approximated by multiple wavelengths) to achieve a wavelength-dependent illumination.However, to find a suitable freeform lens that capable of achieving the optimized LIDC of broader spectra is challenging and is worth investigating in the future.

Fig. 1 .
Fig. 1.A schematic diagram of a common edge-lit LED illuminator designed to provide homogeneous illuminance at the target illumination area.The figure also illustrates an example of dividing the target illumination area into several target zones (T-zones).

Fig. 2 .
Fig. 2. A schematic diagram showing the dividing of the LED radiant intensity distribution into multiple light source zones.The proposed method divides the LED radiant intensity distribution into M source zones (S-zones) according to its light-emitting angle θ to a reference direction n .
suitable choice of each step size, the R value increases with every change of weighting array G, i.e., ( ) ( ) ( )

Fig. 3 .
Fig. 3. Illustration of the steepest descent method used to maximize the R value: (a) the change of the flux weighting array G n is along the R-increasing direction, ( ) n R G ∇ ; (b) the contour lines of the R value of a illumination system.With the suitable choice of each step size of the steepest descent method, the R value increases with every change of flux weighting array G.
be a perfect point light source with the LIDC of the conventional Lambertian form, with the light source being situated at the curvature center of the left side's spherical surface.The arrangement of the light source and the left side's spherical surface avoids considerable Fresnel reflection loss as light enters the lens.The following figure details the design of the freeform surface of the right side of the lens which converts the Lambertian LIDC of a point light source into a specified LIDC, I(θ).

Fig. 4 .
Fig. 4. Illustration of the method for finding a freeform lens to create a specified LIDC: (a) parameters used in the proposed method; (b) calculation of the freeform lens curve.

Fig. 5 .
Fig. 5. Flow chart showing the one-time ray-tracing optimization method process used for designing illumination systems.The symbol ε denotes the residual error of the gradient of R.

Fig. 7 .
Fig. 7. Schematic diagram of the structure of the tube photo-bioreactor illuminator: (a) composition of tube illuminator, with a tube lightguide, several point-like light sources and two tube reflectors at each end of the tube lightguide.The blue rectangle indicates the location of the cross-section plot, as shown in Fig. 7(b); (b) radial cross-section plot of the tube illuminator showing the circular groove structure on both tube lightguide surfaces; (c) transverse cross-section of the tube illuminator showing locations of the inner and outer cylindrical target surfaces in the following optimization settings.The division of two target surfaces is along the direction of cylinder axis (i.e., z-axis).

Fig. 8 .
Fig. 8. Unfolded illuminance distributions of the two cylindrical target surfaces of the initial tube illuminator design: (a) illuminance distribution on inner target surface; (b) illuminance distribution on outer target surface.The color bar of each figure shows the ratio of the illuminance distribution composition.The minimum-to-maximum illuminance ratio R of the two target surfaces was 45.9% and 47.2%, respectively.

Fig. 9 .
Fig. 9. Cross-section average illuminance distributions resulting from each light source zone on the two cylindrical target surfaces: (a) inner target surface and (b) outer target surface.

Fig. 10 .
Fig. 10.Performance of the tube illuminator after LED LIDC optimization: (a) optimized LED LIDC; (b) unfolded illuminance distribution on inner target surface; (c) unfolded illuminance distribution on outer target surface.The color bars of Figs.10(b) and 10(c) indicate the relative composition of the illuminance distributions of the two cylindrical target surfaces.

Fig. 11 .
Fig. 11.Performance of the tube illuminator using freeform lens-constructed light sources: (a) side view of the built freeform lens; (b) LIDC of freeform lens-constructed light source using Lambertian LED chips of different widths (colored lines).The black dotted line denotes the ideal situation, i.e., the optimized LIDC.The symbol r denotes the ratio of the LED chip diameter to the diameter of the spherical side of the lens; (c) unfolded illuminance distribution on inner target surface (while LED chip r = 0.05 mm); (d) unfolded illuminance distribution on outer target surface (while LED chip r = 0.05 mm); (e) cross-section illuminance distributions in the photo-bioreactor in water from the axis of the tube illuminator in six different distances: 10mm, 20mm, 30mm, 60mm, 70mm and 80mm.The color bars of Figs.11(c) and 11(d) indicate the relative composition of the illuminance distributions of the two target surfaces.

4. 1
How to avoid the illumination system performance degrading when using the freeform lens-constructed light source When comparing Figs.10(b) and 10(c) with Figs.11(c) and 11(d), the four figures show there was an obvious decline in the performance of the tube illuminator when the point-like light sources possessing optimized LIDC were replaced by freeform lens-constructed light sources.

Fig. 12 .
Fig. 12. Performance of the tube illuminator using a 10-time-contraction freeform lensconstructed light source rather than the light source used in Fig. 11: (a) unfolded illuminance distribution on inner target surface; (b) unfolded illuminance distribution on outer target surface.The color bars of Figs.12(a) and 12(b) indicate the relative composition of the illuminance distributions.