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
The energy efficiency and compactness of an illumination system are two main concerns in illumination design for extended light sources in practical applications . 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 . 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 θmin≤θ≤θmax (θmax = -θmin = 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
Section 7.5 of  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), C1C2 and F1F2, respectively, represent the initial curves on the entrance profile and the exit profile, and AB denotes the light source. Here, C2 and F2 are mirror points of C1 and F1 about the z-axis, respectively. It is required that the two edge rays, BF2 and AF1, 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) = It(β) and the x-coordinate of point C1 equals W(0)/2 because the luminance of the light source is assumed to be unity. Then, the z-coordinate of point C1, zC, is preset to make sure the incident ray BE1 impinging the initial curve F1F2 at E1 pass through C1 and take the resulting direction angle βN after refraction of the initial curves (note that βN should satisfy the condition that βN<βmax.). Meanwhile, zC 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 F1 and an appropriate value for the first derivative of the curve C1C2 at point C1. These two variables in this design are chosen so that the first derivative of the curve F1F2 at point F1, which is calculated by the condition that the ray 1 propagating inversely is refracted toward A, is negative. That is, F1F2 should be concave downward. With polynomial fitting, we employ two parabolas to represent the two initial curves.
- (2) Calculate a new portion C1PN + i of the exit profile, as shown in Fig. 2(b). Here, PN + i is a point such that the ray BE2 passing through PN + i exits the lens toward direction β = βmax, and the curve Q0QN + i is a portion of the parabola which represents the initial curve of the entrance profile. Take the calculation of point PN + 1 as an example. Set a small value ∆l [for example, ∆l = W(0)/1000], and then the point PN + 1 can be calculated by this equation PN + 1 = PN + ∆l × TN (here, PN is the position vector of the previous point, PN (C1), and TN is the unit tangent vector at point PN.). Find P1 on the portion of the exit profile such that the distance between PN + 1 and the ray 4, the outgoing ray originating from B and exiting the lens at P1, equals W(β1). Then, the normal vector at point PN + 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 PN + i.
- (3) Calculate a new portion PN + iPN1 of the exit profile to ensure those rays originating from B and passing through E2QN + i on the entrance profile be refracted by PN + iPN1 toward the direction β = βmax, as shown in Fig. 2(b).
- (4) Calculate a new portion QN + iQN1 of the entrance profile, as shown in Fig. 2(c). Take the calculation of point QN + 1 + i as an example. Find Pi + 1 on the portion of the exit portion already obtained, which is a point such that the distance between PN + i + 1 and the ray 10, which exits the lens at Pi + 1, equals W(βi + 1). The ray 9, which is parallel to the ray 10, impinges the entrance profile at QN + i + 1, which is the intersection point between the tangent line of its previous point QN + i and the refracted ray. The normal vector at QN + i + 1 can be calculated by Snell’s law. Repeat this calculation till we get to the point PN1.
- (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 PN2 toward direction β = βC (here, βC is the maximum effective angle.). Then, the energy efficiency within the region [-βC, βC] 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 F1 and the first derivative of the curve C1C2 at the point C1, and repeat Steps 1-5 to maximize the energy efficiency within the region [-βC,βC].
Next, an example is given to verify this method and to demonstrate some intrinsic features of this method. The target intensity distribution is defined byEq. (1). We assume that D = 3mm, zC = 6.9mm and the refractive index, n, equals 1.5902. First, we employ only one optical surface here using the method in  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 βC = 20.26°, and the energy efficiency within the region [-βC,βC] 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 C2 to PN2. 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 C1 to PN2. 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 βmax 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 x2 + (z + 2.5)2 = 42. The exit surface is an aspherical surface which is designed using the method in . 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 βC = 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 PN + i to PN2, 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 F1 and the optimized first derivative of the curve C1C2 at the point C1 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 βC = 38.71° and the energy efficiency within the region [-βC,βC] 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 βmax when the impinging point moves from PN + i to PN2, 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 [-βmax,βmax] 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 βmax when the impinging point of this ray on the exit profile moves from PN + i to PN2. 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 PN + iPN2:
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, rms2. A smaller value of rms2 represents less difference (i.e. a better agreement) between the actual intensity and the target. From Fig. 4(a) we have βC = 39.35° and rms2 = 0.0061 within [-βC,βC] (for the intensity distribution obtained from the first method, rms2 = 0.0056.). The energy efficiency within this region equals 97.77%, and the efficiency within the target region [-βmax,βmax] 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, rms1 = 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 asEq. (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 rms2 is small enough (e.g., rms2<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.
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  and the other one is direct design . In our future work, we will generalize the second proposed method to 3D designs for extended non-Lambertian sources.
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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