## Abstract

The energy efficiency and compactness of an illumination system are two main concerns in illumination design for extended sources. In this paper, we present two methods to design compact, ultra efficient aspherical lenses for extended Lambertian sources in two-dimensional geometry. The light rays are directed by using two aspherical surfaces in the first method and one aspherical surface along with an optimized parabola in the second method. The principles and procedures of each design method are introduced in detail. Three examples are presented to demonstrate the effectiveness of these two methods in terms of performance and capacity in designing compact, ultra efficient aspherical lenses. The comparisons made between the two proposed methods indicate that the second method is much simpler and easier to be implemented, and has an excellent extensibility to three-dimensional designs.

© 2016 Optical Society of America

## 1. Introduction

The energy efficiency and compactness of an illumination system are two main concerns in illumination design for extended light sources in practical applications [1]. Since the étendue of an extended source (an actual source) is nonzero, usually those zero-étendue algorithms with the assumption of ideal source will be invalid in compact designs [2–8]. A number of algorithms which have been developed for extended sources by taking the étendue of an extended source into account [9–15]. Although those algorithms for extended sources have a potential to achieve a significant reduction in size of the illumination system along with a good compactness and high energy efficiency, the illumination design for extended sources is still not well addressed and faces many unresolved challenges. Most of the existing algorithms for extended sources employ a single aspherical surface to redistribute the spatial energy of a light source [9–15]. Due to the limitation of a single aspherical surface, the controls on the distribution of light rays are limited, and we usually have to make a trade-off between the compactness and energy efficiency [13–15] (see Fig. 1). Fortunately, this limitation of one single surface can be overcome by using two optical surfaces [1]. However, the illumination design of two optical surfaces is more challenging than that of one single surface, and this type of two-surface design is still an open problem.

In this paper, we address the two-surface design problem and present two methods to design compact, ultra efficient aspherical lenses for extended Lambertian sources. Since two-dimensional (2D) designs are in any case the first step towards a generalization for three-dimensional (3D) designs, we focus on the design of two refractive surfaces in 2D geometry. Compared to our previous publications [13–15], the contribution of this paper is threefold: we first reveal an intrinsic feature of the compact and ultra efficient two-surface design; secondly, the new methods presented in this paper overcome the limitations of one single surface design in our previous work [13–15], a good compactness along with high energy efficiency is easily achieved; thirdly, the second proposed method is much simpler and easier to be implemented, and has an excellent extensibility to 3D designs.

## 2. Calculation of two aspherical surfaces and an intrinsic feature

Since a prescribed illuminance design can be converted into a prescribed intensity design as long as the influence of the lens size on the performance can be ignored, we only address the prescribed intensity design in this paper. We also assume the extended source is a Lambertian line source with a length of D and an angular range of emission between θ_{min}≤θ≤θ_{max} (θ_{max} = -θ_{min} = 90°). The line source considered here is such a light source whose spatial energy distribution is spatially confined to a plane containing the line source and the optical axis. Since the luminance of a Lambertian source is constant, we further assume the luminance of the source equals unity. The conservation law of energy says the total flux of the outgoing beam should be equal to that of the incident beam in a loss-less system, which can be rewritten as

_{t}(β) is the prescribed output intensity distribution with an angular range between β

_{min}≤β≤β

_{max}(here, β

_{min}= -β

_{max}). Thus, our design goal is to achieve a good compactness of the aspherical lens along with a high energy efficiency within the region [β

_{min},β

_{max}]. Here, the energy efficiency of the aspherical lens is defined as the ratio of the energy refracted into the desired region to the energy of the light source.

Section 7.5 of [1] described a method which could be used to maximize the energy efficiency of an illumination system by using two aspherical surfaces. Unfortunately, the feasibility of that method described therein has not been proved yet and its implementation still needs to be further explored. Inspired by this method, we develop the first method which uses two aspherical surfaces to achieve a compact ultra efficient lens. Due to the nature of an extended source, two initial curves are required before we start to design the lens profiles. In Fig. 2(a), C_{1}C_{2} and F_{1}F_{2}, respectively, represent the initial curves on the entrance profile and the exit profile, and AB denotes the light source. Here, C_{2} and F_{2} are mirror points of C_{1} and F_{1} about the z-axis, respectively. It is required that the two edge rays, BF_{2} and AF_{1}, should be refracted by the initial curves to take the resulting direction angle β = 0°. That is, the two outgoing rays, 1 and 2, are parallel to the z-axis. Let W(β) represent the distance between two parallel outgoing rays which originate from A and B, respectively. We have that W(β) = I_{t}(β) and the x-coordinate of point C_{1} equals W(0)/2 because the luminance of the light source is assumed to be unity. Then, the z-coordinate of point C_{1}, z_{C}, is preset to make sure the incident ray BE_{1} impinging the initial curve F_{1}F_{2} at E_{1} pass through C_{1} and take the resulting direction angle β_{N} after refraction of the initial curves (note that β_{N} should satisfy the condition that β_{N}<β_{max}.). Meanwhile, z_{C} should also meet the prescribed requirements on the lens size.

Then, the design of the two profiles follows the following six steps.

- (1) Choose an appropriate value for the z-coordinate at F
_{1}and an appropriate value for the first derivative of the curve C_{1}C_{2}at point C_{1}. These two variables in this design are chosen so that the first derivative of the curve F_{1}F_{2}at point F_{1}, which is calculated by the condition that the ray 1 propagating inversely is refracted toward A, is negative. That is, F_{1}F_{2}should be concave downward. With polynomial fitting, we employ two parabolas to represent the two initial curves. - (2) Calculate a new portion C
_{1}P_{N + i}of the exit profile, as shown in Fig. 2(b). Here, P_{N + i}is a point such that the ray BE_{2}passing through P_{N + i}exits the lens toward direction β = β_{max}, and the curve Q_{0}Q_{N + i}is a portion of the parabola which represents the initial curve of the entrance profile. Take the calculation of point P_{N + 1}as an example. Set a small value ∆l [for example, ∆l = W(0)/1000], and then the point P_{N + 1}can be calculated by this equation**P**_{N + 1}=**P**_{N}+ ∆l ×**T**_{N}(here,**P**_{N}is the position vector of the previous point, P_{N}(C_{1}), and**T**_{N}is the unit tangent vector at point P_{N}.). Find P_{1}on the portion of the exit profile such that the distance between P_{N + 1}and the ray 4, the outgoing ray originating from B and exiting the lens at P_{1}, equals W(β_{1}). Then, the normal vector at point P_{N + 1}can be calculated by the condition that the ray 5, which is parallel to the ray 4, is refracted toward A. Repeat this calculation till we get to the point P_{N + i}. - (3) Calculate a new portion P
_{N + i}P_{N1}of the exit profile to ensure those rays originating from B and passing through E_{2}Q_{N + i}on the entrance profile be refracted by P_{N + i}P_{N1}toward the direction β = β_{max}, as shown in Fig. 2(b). - (4) Calculate a new portion Q
_{N + i}Q_{N1}of the entrance profile, as shown in Fig. 2(c). Take the calculation of point Q_{N + 1 + i}as an example. Find P_{i + 1}on the portion of the exit portion already obtained, which is a point such that the distance between P_{N + i + 1}and the ray 10, which exits the lens at P_{i + 1}, equals W(β_{i + 1}). The ray 9, which is parallel to the ray 10, impinges the entrance profile at Q_{N + i + 1}, which is the intersection point between the tangent line of its previous point Q_{N + i}and the refracted ray. The normal vector at Q_{N + i + 1}can be calculated by Snell’s law. Repeat this calculation till we get to the point P_{N1}. - (5) Repeat Steps 3 and 4 until the calculation of the data points cannot converge (This will be discussed later in this paper). The ray 11 originating from the point A exits the lens at the end point of the exit profile P
_{N2}toward direction β = β_{C}(here, β_{C}is the maximum effective angle.). Then, the energy efficiency within the region [-β_{C}, β_{C}] is defined as the ratio of the energy within this region to the total energy of the light source. - (6) Optimize the z-coordinate of the point F
_{1}and the first derivative of the curve C_{1}C_{2}at the point C_{1}, and repeat Steps 1-5 to maximize the energy efficiency within the region [-β_{C},β_{C}].

Next, an example is given to verify this method and to demonstrate some intrinsic features of this method. The target intensity distribution is defined by

_{1}is a constant which can be calculated by applying the energy conservation given in Eq. (1). We assume that D = 3mm, z

_{C}= 6.9mm and the refractive index, n, equals 1.5902. First, we employ only one optical surface here using the method in [13] and let the light source be immersed in the lens. The actual intensity is given in Fig. 3(a). From this figure we can see that the maximum effective angle β

_{C}= 20.26°, and the energy efficiency within the region [-β

_{C},β

_{C}] only equals 43.99%. In Fig. 3(b), the blue solid line denotes the change of the direction angle of an outgoing ray which originate from B when the impinging point of this ray on the exit profile moves from C

_{2}to P

_{N2}. The red dashed line denotes the change of the direction angle of an outgoing ray originating from A when the impinging point of this ray on the exit profile moves from C

_{1}to P

_{N2}. The abscissa represents the polar angle of the impinging point from the coordinate origin. From this figure we can see that a lot of rays originating from B are not well controlled due to the limitation of one single surface. Besides, we observe that the calculation of the lens profile cannot converge onto the x-axis, as shown in Fig. 3(c). That is because the target angle β

_{max}is very close to the critical angle 38.97° so that the total internal reflection takes place when the lens profile approaches the x-axis.

As a second example to achieve the target design with two surfaces, the entrance surface is prefixed as a spherical surface, for example the surface defined as x^{2} + (z + 2.5)^{2} = 4^{2}. The exit surface is an aspherical surface which is designed using the method in [13]. The lens profile is given in Fig. 3(c), and the actual intensity distribution is given in Fig. 3(a). From the actual intensity distribution denoted by the red dashed line in Fig. 3(a), we observe β_{C} = 34.34° and the energy efficiency within this region equals 81.37%. This design is obviously better than the previous one due to the application of two surfaces; however, the light rays still are not controlled well when the impinging point moves from P_{N + i} to P_{N2}, as shown in Fig. 3(d).

Next, two aspherical surfaces are used here to direct the light rays and the first method presented above is applied to the design of the two aspherical surfaces. The optimized z-coordinate of the point F_{1} and the optimized first derivative of the curve C_{1}C_{2} at the point C_{1} are given in Table 1. The actual intensity distribution is depicted in Fig. 4(a) and the lens profile is given in Fig. 4(b). The maximum effective angle β_{C} = 38.71° and the energy efficiency within the region [-β_{C},β_{C}] equals 95.54%. One intrinsic feature of this method is that the light rays originating from B are controlled very well with the direction angle equaling β_{max} when the impinging point moves from P_{N + i} to P_{N2}, as shown in Fig. 4(c). Although this is an intrinsic feature of 2D designs, this feature can be generalized to 3D rotational designs and even freeform designs. Because of this feature, the energy efficiency within the target region [-β_{max},β_{max}] can be very high, 99.35% in this design. Let H denote the z-coordinate of the vertex of the exit profile. From Fig. 4(b) we have that the ratio H/D = 2.34. Obviously, a good compactness and high energy efficiency are simultaneously achieved in this design by using the first method. However, we need to point out that the calculation of the entrance profile usually cannot converge onto the x-axis, as shown in Fig. 4(b). That is because we only employ the parabola to represent the initial curves and the degrees of design freedom of the parabola are limited. The entrance profile may converge onto the x-axis by using higher-order polynomials; however, it is not necessary because those rays which are not captured by the lens account for only 0.44% of the total energy and the application of the parabolas in the first method can significantly simplify the numerical optimization process. Besides, since two profiles need to be calculated, this method may not have a good extensibility to 3D designs.

## 3. Calculation of an aspherical surface and an optimized parabola

Figures 3(b) and 3(d) reveal a fact that the actual direction angle of the outgoing ray which originates from B deviates from the target angle β_{max} when the impinging point of this ray on the exit profile moves from P_{N + i} to P_{N2}. The deviation determines the optical performance. Here, we employ the fractional RMS to quantify the difference between the actual direction angle and the target one on the portion P_{N + i}P_{N2}:

_{1}is the number of the sample points, and β

_{ak}is the actual direction angle at the k-th point. The optical performance of the illumination design of two optical surfaces can be improved by reducing the difference between the actual direction angle and the target one. Based on these analyses and the first method, we propose the second design method for designing compact, ultra efficient aspherical lenses for extended Lambertian sources, which is much simpler and easier to be implemented than the first method. Two optical surfaces are also used in the second method: the exit surface is an aspherical surface and the entrance surface is an optimized parabola. The design of these two surfaces follows the next three steps. (i) Use same Step 1 as the first method to find two parabolas: one parabola is used to represent the initial curve of the exit profile and the other one is used as the entrance profile. (ii) Apply same Step 2 as the first method to the calculation of the rest of the exit profile. (iii) Optimize the two variables and repeat Steps (i) and (ii) to minimize

*rms*

_{1}.

We have also applied the second proposed method to the first example. The actual intensity distribution is given in Fig. 4(a). The difference between the actual intensity and the target is also quantified by the fractional RMS, *rms*_{2}. A smaller value of *rms*_{2} represents less difference (i.e. a better agreement) between the actual intensity and the target. From Fig. 4(a) we have β_{C} = 39.35° and *rms*_{2} = 0.0061 within [-β_{C},β_{C}] (for the intensity distribution obtained from the first method, *rms*_{2} = 0.0056.). The energy efficiency within this region equals 97.77%, and the efficiency within the target region [-β_{max},β_{max}] equals 99.69%. The lens profile depicted in Fig. 4(b) tells us that the lens is very compact with the ratio H/D being 2.34. Besides, *rms*_{1} = 0.0011, which means the light rays are controlled very well with little difference between the actual direction angle and βmax, as illustrated by the blue solid line in Fig. 4(d). From Fig. 4 we can see there is little difference between the actual design and the target, and the performance of the two lenses designed by the first method and the second method are excellent and quite similar.

Further, we also apply the two proposed methods to another two design examples with the intensity distribution defined as

_{2}and K

_{3}can be calculated by Eq. (1). The design parameters are given in Table 1 and the results of these two examples are given in Table 2 and Fig. 5. From these results we can also observe that the designs achieved by these two methods are excellent and quite similar, which means we can use one of the two methods to achieve a compact ultra efficient design. It is also of interest to mention that usually it is not necessary to find the best optimization solution, because there will be little difference in optical performance between different solutions when rms

_{2}is small enough (e.g., rms

_{2}<0.01). According to the procedures of these two design methods described above, we can find that the second method is much simpler and easier to be implemented than the first method. Besides, it is also of interest to mention that the second method may have an excellent extensibility to 3D designs for extended sources, especially for extended non-Lambertian sources, because the entrance surface used in the second method is a parabola, which is a very simple aspherical surface.

## 4. Conclusion

In conclusion, we present two methods to design compact and ultra efficient aspherical lenses for extended Lambertian sources in 2D geometry. Due to the limitation of one single surface, two optical surfaces are employed in the two proposed methods. The examples clearly show that the designs achieved by the use of the two proposed methods are excellent and quite similar. The two proposed methods have an effective control of glare because the light beams from the source are controlled very well. Besides, since the calculation of the lens profile in the two methods is essentially a numerical optimization with only two variables, the convergence of the two methods can be very fast and stable. Since the entrance surface in the second method is a parabola, the second method is much simpler and easier to be implemented, and has an excellent extensibility to 3D designs. We have shown in our previous work that there are at least two ways to generalize 2D designs to 3D designs: one is illuminance feedback [13] and the other one is direct design [15]. In our future work, we will generalize the second proposed method to 3D designs for extended non-Lambertian sources.

## Acknowledgment

This work was partially supported by the National Institute of Biomedical Imaging and Bioengineering (grant no. 1R01EB18921) and the National Cancer institute (grant no. R01CA171651).

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