## Abstract

Achieving accurate control of the spatial energy distribution of extended sources is an ultimate goal of illumination design and has considerable significance in practical applications. To achieve this goal, we, for the first time, present a direct method to design compact and ultra-efficient freeform lenses for extended sources in three-dimensional geometries. Instead of using the traditional Monte Carlo ray-tracing and feedback methods to address the challenges caused by the extended sources, we develop a novel design method to numerically calculate both surfaces of the freeform lens directly. The proposed method is very efficient and robust in designing freeform lenses for Lambertian and non-Lambertian extended light sources. The final freeform lenses are very compact and highly efficient and can significantly reduce glare and light trespass.

© 2016 Optical Society of America

Freeform optics design is an important technique widely used in optical systems to redistribute the spatial energy distribution of a light source [1]. Most of the currently available design methods start with an ideal point source. However, in practical applications, an actual source has a certain spatial and angular extent. Consequently, those zero-étendue algorithms with the assumption of a point source usually become invalid [2–6]. Taking the spatial and angular extent of an extended source into account, freeform lens design in three-dimensional (3D) geometry is much more difficult because an extended source is an infinite number of point sources. It is impossible to achieve good control on the spatial energy distributions of all the point sources using the freeform lens designed for a single point source. Inevitably, the problems of glare, light trespass, over-illumination, and energy waste will arise, and trade-offs must be made between the optical performance and compactness of a system. Traditional optimization methods are sometimes used to design freeform surfaces for extended sources [7,8]. Due to the nature of optimization, the design result is strongly determined by the starting point, which is usually obtained by a zero-étendue algorithm, and it will be difficult to optimize the freeform optics without suitable surface parameterization. Besides, the limited number of variables and the cumbersome Monte Carlo ray tracing are the other two major limitations of the traditional optimization methods. A similar technique to optimization design is feedback design, which iteratively uses illuminance/intensity compensation to improve the performance of a design created by a zero-étendue method [9,10] with cumbersome Monte Carlo ray tracing. Since the feedback design is similar to a local optimization, the result is also strongly determined by the starting point. Besides, the feedback methods are less effective in compact designs [11]. For both traditional optimization and feedback methods, trade-offs must be made between the design efficiency and the optical performance of the system. To our knowledge, the freeform lens design for extended sources in 3D geometry is still not well addressed and faces many unresolved challenges.

In this Letter, we address the freeform lens design for extended sources by developing a direct design method to achieve accurate control of the spatial energy distribution of extended sources. There are three main contributions of this work: (1) the proposed method overcomes the limitations of the existing methods, achieving compact and highly efficient systems, and glare and light trespass are significantly reduced; (2) the profiles of the freeform lens are numerically calculated by the proposed method without cumbersome Monte Carlo ray tracing, which may open up a new avenue for the freeform optics design for extended sources; and (3) the proposed method is applicable to Lambertian/non-Lambertian disk sources and LED sources.

Assume that the source is a non-Lambertian disk source, and its luminance is $L(r,\varphi )$, where $r$ is the radius of a point where the incident ray emanates from the source, and $\varphi $ is the angle between the ray and the optical axis, which is perpendicular to the source at the center of the source. When $L(r,\varphi )$ is both direction invariant and space invariant, the non-Lambertian source degenerates into a Lambertian source. Note that, if the source is a square or rectangular LED emitter, we can use a Lambertian disk source with the same area to replace the emitter in the design to keep the total output of the source unchanged. With $L(r,\varphi )$, the total flux of the source in the air is given by

A freeform lens is used in our method to redistribute the spatial energy distribution of the extended source. The design strategy given in Fig. 1 shows that there are two design phases in this method. The purpose of the first phase is to find an optimal entrance surface of the freeform lens. Instead of designing the entrance surface as an unconstrained freeform surface, a simple paraboloid surface is selected in our method to simply the design. A higher-order polynomial is not necessary because the paraboloid surface can meet the design requirements and significantly simplify the design. With the optimized entrance surface, we then do a direct 3D design for the extended source in the second phase. More details about the proposed method are introduced below.

To perform a two-dimensional (2D) design in the first phase, we need to calculate the target output intensity ${I}_{2\mathrm{D}}(\beta )$ for the 2D design. In a 3D geometry, the illuminance $E(r)$ produced by an ideal point source is given by

where ${K}_{1}$ is a constant. We assume that we need to produce the same illuminance $E(r)$ along the $x$ axis in the 2D geometry. As shown in Fig. 2, the target intensity ${I}_{2\mathrm{D}}(\beta )$ should satisfyThe line source considered in the 2D design is such a source whose spatial energy distribution is spatially confined to a plane containing the source and the optical axis. The 2D design is also confined to this plane, which is a meridional plane of the extended source. Since the luminance distribution of a non-Lambertian line source is a function of the position and direction, both the edge rays and interior rays from the source need to be considered in the design. For a Lambertian line source, however, only the edge rays from the source are considered [1]. Thus, the design of the freeform lens for a non-Lambertian line source will be quite different from that for a Lambertian line source and will be much more difficult.

Assuming that the entrance surface of the freeform lens in the first design phase is a parabola defined as $z={g}_{1}{x}^{2}+{g}_{0}$, we choose appropriate values for ${g}_{0}$ and $x$ coordinate ${X}_{0}$ of the end point ${Q}_{M}$ of the entrance profile, which are two independent variables, to make sure the entrance profile is concave downward, as shown in Fig. 3(a). Then, ${g}_{1}$ can be calculated as ${g}_{1}=-{g}_{0}/{X}_{0}^{2}$. As shown below, the calculation of the exit surface of the freeform lens is based on the entrance surface, which means the performance of the lens is fully determined by the entrance surface. Thus, finding appropriate values for ${g}_{0}$ and ${X}_{0}$ is critical. Here, we develop an algorithm to optimize ${g}_{0}$ and ${X}_{0}$ to achieve accurate control of the light distribution of the non-Lambertian line source. The calculation of the lens profiles in the first design phase is converted to a 2D numerical optimization, which means the 2D design can converge stably and quickly. In Fig. 3(a), ${P}_{0}$ is a mirror point of ${P}_{N}$ about the $z$ axis. It is required that the two outgoing edge rays 1 and 2 be parallel to the $z$ axis. The $z$ coordinate of ${P}_{N}$, ${z}_{PN}$, is preset to make sure the incident ray ${\mathrm{BE}}_{1}$ exits the lens toward the direction $\beta ={\beta}_{N}$ (${\beta}_{N}<{\beta}_{\mathrm{max}}$). To find an initial curve ${P}_{0}{P}_{N}$ on the exit surface, we choose a value for the $x$ coordinate of ${P}_{N}$ and calculate the first derivative of the curve ${P}_{0}{P}_{N}$ at ${P}_{N}$ by the fact that ray 2 coming from the end point A of the line source is parallel to the $z$ axis. We then employ a parabola to represent the curve ${P}_{0}{P}_{N}$ and calculate a set of outgoing rays between the rays 1 and 2, which are refracted toward the direction $\beta =0\xb0$. Further, we can also get the luminance of these outgoing rays. The output intensity in the direction $\beta =0\xb0$ is the integral of the luminance function of all the outgoing rays at this direction. Then, we just need to change the $x$ coordinate of ${P}_{N}$ to change the luminance of those outgoing rays between the rays 1 and 2 so the condition in which ${I}_{a}(0)={I}_{2\mathrm{D}}(0)$ is satisfied, where ${I}_{a}(0)$ is the actual output intensity at this direction.

To obtain the entire exit profile, as shown in Fig. 3(b), using point ${P}_{N+1}$ as an example, we first trace a ray from the end point B of the source, which impinges on the exit profile at ${P}_{1}$, and then calculate the direction angle ${\beta}_{1}$ of ray 4. For a small value of $\mathrm{\Delta}l$, we can calculate the new point ${P}_{N+1}$ as ${\mathbf{P}}_{N+1}={\mathbf{P}}_{N}+\mathrm{\Delta}l\times {\mathbf{T}}_{N}$ (here, ${\mathbf{P}}_{N}$ is the position vector of the point ${P}_{N}$, and ${\mathbf{T}}_{N}$ is the unit tangent vector at ${P}_{N}$). Assuming a ray from A impinges the exit profile at ${P}_{N+1}$ and exits toward the direction $\beta ={\beta}_{1}$, we can calculate the normal vector of the exit profile at ${P}_{N+1}$ by Snell’s law. Since ${P}_{1}{P}_{N+1}$ is already known, we can get the luminance of those outgoing rays refracted by ${P}_{1}{P}_{N+1}$ toward the direction $\beta ={\beta}_{1}$. Similarly, the intensity in this direction is the integral of the luminance of all the outgoing rays at this direction. The next step is to change $\mathrm{\Delta}l$ so that the output intensity in this direction equals the target intensity ${I}_{2\mathrm{D}}({\beta}_{1})$. Then, we repeat the whole calculation described in this step until the incident ray from A converges onto the $x$ axis.

With the obtained 2D profile, the distribution of the direction angles of the outgoing rays originating from A and B is estimated in Fig. 3(c). The fractional root mean square (RMS) is employed here to quantify the difference between the actual angle and the target angle on ${P}_{N1}{P}_{M}$, which is defined as

*Num*is the number of sample points, and ${\beta}_{ak}$ is the actual direction angle at the $k$th point. Figure 3(c) indicates that the outgoing rays will be better controlled by reducing the difference between the actual direction angle and ${\beta}_{\mathrm{max}}$. Thus, we need to optimize ${g}_{0}$ and the $x$ coordinate ${X}_{0}$ of ${Q}_{M}$ and repeat steps (1) and (2) to minimize ${\mathrm{rms}}_{1}$.

Once we finalize the 2D profile of the entrance surface, the entrance surface can be obtained by rotating the optimized parabola. In the 3D design, the intensity of the outgoing rays along a given direction [e.g., ($\mathrm{sin}\text{\hspace{0.17em}}{\beta}_{i}$, 0, $\mathrm{cos}\text{\hspace{0.17em}}{\beta}_{i}$)] is the integral of the luminance function of these outgoing rays,

In Fig. 4(b), the coordinate plane $xOz$ is a meridional plane of the lens, and the curve ${C}_{0}{C}_{N}$ is located in this plane. The $z$ coordinate of ${C}_{N}$, ${z}_{C}$, is equal to that of ${P}_{N}$ in Fig. 3(a) and ${C}_{0}$ is a mirror point of ${C}_{N}$ about the $z$ axis. The meridian ray from $A$, impinging on the entrance and exit surfaces at ${D}_{N}$ and ${C}_{N}$, exits the lens parallel to the $z$ axis. In order to calculate the first derivative of ${C}_{0}{C}_{N}$ at ${C}_{N}$, it is necessary to find ${D}_{N}$ on the entrance surface, which is a paraboloid surface designed in the first phase. The optical path length (OPL) of this ray between $A$ and ${C}_{N}$ can be written as $\mathrm{OPL}={l}_{1}+n{l}_{2}$, where ${l}_{1}$ and ${l}_{2}$ are the lengths of $A{D}_{N}$ and ${D}_{N}{C}_{N}$, respectively, and $n$ is the refractive index of the lens. Fermat’s principle states that the OPL between $A$ and ${C}_{N}$ is an extremum along the ray containing these points. Thus, ${D}_{N}$ should satisfy the condition that $\partial (\mathrm{OPL})/\partial {x}_{D}=0$, where ${x}_{D}$ is the $x$ coordinate of ${D}_{N}$. When ${D}_{N}$ is obtained, the first derivative of ${C}_{0}{C}_{N}$ at ${C}_{N}$ can be derived. With polynomial fitting, we employ a parabola to represent ${C}_{0}{C}_{N}$ and further rotate ${C}_{0}{C}_{N}$ around the $z$ axis to get the initial patch, which is a paraboloid surface. Since the entrance surface is also a paraboloid surface, it is not difficult to calculate those incident rays from the source which exit the lens toward the direction (0, 0, 1). Using Eq. (6), the actual output intensity ${I}_{a}(0)$ in this direction can be obtained. The next step is to adjust the $x$ coordinate of point ${C}_{N}$ to satisfy the condition that ${I}_{a}(0)={I}_{3\mathrm{D}}(0)$.

When the initial patch is ready, the rest of the lens profile can be calculated accordingly. Using ${C}_{N+1}$ as an example, we first trace a meridian ray from $B$, which impinges on the exit profile at ${C}_{1}$, and then calculate the direction ($\mathrm{sin}\text{\hspace{0.17em}}{\beta}_{1}$, 0, $\mathrm{cos}\text{\hspace{0.17em}}{\beta}_{1}$) of its outgoing ray, as shown in Fig. 4(c). For a small value of $\mathrm{\Delta}{l}_{1}$, we can calculate the new point ${\mathbf{C}}_{N+1}$ as ${\mathbf{C}}_{N+1}={\mathbf{C}}_{N}+\mathrm{\Delta}{l}_{1}\times {\mathbf{T}}_{CN}$ (here, ${\mathbf{C}}_{N}$ is the position vector of point ${C}_{N}$, and ${\mathbf{T}}_{CN}$ is the unit tangent vector at ${C}_{N}$ in the meridional plane). Assuming a meridian ray from A impinges the exit surface at ${C}_{N+1}$ and exits toward the direction ($\mathrm{sin}\text{\hspace{0.17em}}{\beta}_{1}$, 0, $\mathrm{cos}\text{\hspace{0.17em}}{\beta}_{1}$), the normal vector of the exit profile at ${C}_{N+1}$ can be calculated by Snell’s law. With the known portion $V{C}_{N+1}$ ($V$ is the vertex of the exit surface), a surface patch can be constructed by rotating $V{C}_{N+1}$. Then, we need to calculate those incident rays (meridian rays and skew rays) from the extended source, which exits the surface patch toward the direction ($\mathrm{sin}\text{\hspace{0.17em}}{\beta}_{1}$, 0, $\mathrm{cos}\text{\hspace{0.17em}}{\beta}_{1}$). The direct design of a freeform lens in a 3D geometry requires significant innovation in the mathematical process to simultaneously control both the meridian rays and skew rays of an extended source. We developed an algorithm based on the bisection method in our previous work [12] to calculate the meridian rays and skew rays of an extended source. This algorithm is generalized and applied here to design the compact and ultra-efficient freeform lens. When those incident rays are obtained, the actual intensity ${I}_{a}({\beta}_{1})$ formed by the corresponding outgoing rays at the direction ($\mathrm{sin}\text{\hspace{0.17em}}{\beta}_{1}$, 0, $\mathrm{cos}\text{\hspace{0.17em}}{\beta}_{1}$) is calculated using Eq. (6). Similarly, the next step is to adjust $\mathrm{\Delta}{l}_{1}$ to change the coordinates of ${C}_{N+1}$ so that the condition that ${I}_{a}({\beta}_{1})={I}_{3\mathrm{D}}({\beta}_{1})$ is satisfied. Then, we repeat the whole calculation until the incident ray from $A$ converges onto the $x$ axis.

As an example, the luminance of the non-Lambertian disk source is a function of the position and direction, which is given by

As mentioned above, the source can be a Lambertian square or rectangular source. This feature makes the proposed method applicable to LED lighting. To demonstrate this important feature, we also design a freeform lens to redistribute the spatial energy distribution of an LED source (LUXEON Z ES LXZ2-5070 [13]) to produce the uniform illumination defined by Eq. (8). The size of the LED emitter is $1.5\text{\hspace{0.17em}}\mathrm{mm}\times 1.5\text{\hspace{0.17em}}\mathrm{mm}$. To apply the proposed method, we use a Lambertian disk source with the same area to replace the LED emitter (the diameter of the disk source equals 1.6926 mm). With the proposed method, we get an optimized entrance surface as $z=-0.0728$ $({x}^{2}+{y}^{2})+0.7$ for the parameters ${z}_{C}=5.6\text{\hspace{0.17em}}\mathrm{mm}$ and $n=1.6489$. The intensity produced by the lens with the disk source is represented by the red solid line in Fig. 5(c). From this figure, we have ${\mathrm{rms}}_{2}=0.0056$ and ${\eta}_{1}=95.84\%$ within the region $[0,{\beta}_{C}]$ (${\beta}_{C}=39.42\xb0$) and ${\eta}_{2}=98.97\%$ within the region $[0,{\beta}_{\mathrm{max}}]$, showing the light rays are controlled very well. The lens profiles are denoted by the red solid line in Fig. 5(b) with the ratio $H/D$ of 3.34. Again, the lens is compact and highly efficient. Figure 5(d) gives the illumination patterns on an observation plane at 222 mm; the normalized illuminance distributions along the line $y=0\text{\hspace{0.17em}}\mathrm{mm}$ are also plotted. From Figs. 5(c) and 5(d), we see that the differences between the outputs produced by the two different sources are neglectable. Figure 5(d) also shows that the illuminance slowly drops a little bit from the center to the edge of the pattern, which means the lens does not produce a perfectly uniform illuminance distribution. This is because the influence of the spatial extent of the lens on the optical performance cannot be neglected at the current lighting distance. This influence can, of course, be reduced by increasing the lighting distance (e.g., the influence is neglectable when the distance increases to 1000 mm). Another fact shown in Fig. 5(d) is that the illuminance declines steeply to zero at the edge of the pattern, which again indicates the outgoing rays are controlled very well and are strongly confined to the target region.

The experimental setup and the prototype of the freeform lens are shown in Fig. 6(a); the distance from the LED to the observation plane equals 222 mm. The camera is placed behind the observation plane to record the illumination pattern on the observation plane. The measured illuminance distributions along the lines $y=0\text{\hspace{0.17em}}\mathrm{mm}$ and $x=0\text{\hspace{0.17em}}\mathrm{mm}$ are plotted in Fig. 6(c), showing good agreement between the experimental and the simulated results. The chromatic aberration, fabrication errors, and alignment errors are the main error sources that contribute to minor differences in the edges. The fractional RMS equals 0.0233 within the target region [$-186.28\text{\hspace{0.17em}}\mathrm{mm}$, 186.28 mm], showing the effectiveness of the proposed method.

In conclusion, we present a direct method to design compact and ultra-efficient freeform lenses to achieve accurate control of the spatial energy distribution of extended sources in a 3D geometry. Monte Carlo ray tracing is not needed, and both surfaces of the lens are numerically calculated. The proposed method is applicable to Lambertian/non-Lambertian disk sources and LED sources and has considerable significance in practical applications.

## REFERENCES

**1. **R. Winston, J. C. Miñano, and P. Benítez, *Nonimaging Optics* (Elsevier, 2005).

**2. **H. Ries and J. Muschaweck, J. Opt. Soc. Am. A **19**, 590 (2002). [CrossRef]

**3. **Y. Ding, X. Liu, Z. Zheng, and P. Gu, Opt. Express **16**, 12958 (2008). [CrossRef]

**4. **F. R. Fournier, W. J. Cassarly, and J. P. Rolland, Opt. Express **18**, 5295 (2010). [CrossRef]

**5. **A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, Opt. Express **21**, 10563 (2013). [CrossRef]

**6. **R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, Opt. Lett. **38**, 229 (2013). [CrossRef]

**7. **W. J. Cassarly and M. J. Hayford, Proc. SPIE **4832**, 258 (2002). [CrossRef]

**8. **S. Hu, K. Du, T. Mei, L. Wan, and N. Zhu, Opt. Express **23**, 20350 (2015). [CrossRef]

**9. **J. Bortz and N. Shatz, Proc. SPIE **6338**, 633805 (2006). [CrossRef]

**10. **Y. Luo, Z. Feng, Y. Han, and H. Li, Opt. Express **18**, 9055 (2010). [CrossRef]

**11. **K. Wang, Y. J. Han, H. T. Li, and Y. Luo, Opt. Express **21**, 19750 (2013). [CrossRef]

**12. **R. Wu and H. Hua, Opt. Express **24**, 1017 (2016). [CrossRef]