## 1. Introduction

The generation of required irradiance distribution in a rectangle area is one of the key problems arising in the design of street, home, and industrial lighting systems. This problem can be efficiently solved by using LED light sources with special secondary optics that redistribute the light flux and generate the required irradiance distribution. The problem of secondary optic design consists in the computation of optical surface shapes. The solution to this problem results in the solving of nonlinear differential equations in partial derivatives, similar to the Monge–Ampere equation [1–3]. In general, a 3D case solution of this equation is a challenging problem. Analytical solutions are known only for radially symmetric or cylindrical surfaces [4–7]. Also, analytical methods are proposed for a design of the optical elements for generating line-shaped directivity diagrams [8–10].

For the illumination of streets, roads, or corridors, freeform optical elements producing prescribed irradiance distribution in an elongated rectangular region are required. An elongated rectangular region means a high aspect ratio of rectangle sides (3 or more); usually, the longitudinal angular size of a rectangle is about 120°–150°, and the transversal angular size is less than 60°–70°. To design freeform optical elements that produce desired irradiance distribution in a rectangular region, iterative optimization procedures are used [11–18]. Methods reported in [11–18] are intended for the design of optical elements with a single free-form refractive surface. Usage of such optical elements for generating required irradiance distribution in the mentioned elongated rectangular region leads to essential losses of light efficiency. The reason is that the refraction on the optical surface cannot be used for large-angle ray rotation because of the total internal reflection. “Ray rotation” means the change of the ray propagation direction after the refraction on the optical surface. For example, the maximal angle of ray rotation at the material–air interface for material refraction index 1.5 is less than 49°. This means that the generation of an elongated rectangular irradiance distribution with a low transversal angular size is impossible without significant light efficiency losses. Indeed, at a transversal angular size of 30°–40°, the maximal required angle of ray rotation is 70°–75°. Therefore, a part of the rays cannot be redirected into the illuminated area by refraction.

The design of a high-light-efficient optical element for generating a line-shaped directivity diagram with a large angular size was proposed in [19]. The proposed design comprises two groups of working surfaces. The first group contains three cylindrical surfaces and serves to transform the spherical beam from the light source to a cylindrical beam. The second group is represented by a cylindrical external surface and converts the cylindrical incident beam into a line-shaped directivity diagram. The high light efficiency is achieved by the use of the total internal reflection effect while forming the cylindrical beam. Also, the modified version of the design intended to generate a rectangle-shaped directivity diagram was suggested in [19]. It should be noted that when generating the rectangular directivity diagram, the solution in [19] is approximate. In a general, non-paraxial case, the rectangular directivity diagram (i.e., rectangular intensity distribution in angular coordinates) does not provide specified rectangular irradiance distribution in the plane. Thus, the solution in [19] is not suitable for generating the rectangular-shaped irradiance distribution.

In this paper, we propose to use the optical element in [19] with a free-form external surface for generating the required irradiance distribution in the elongated rectangular region. Such a surface is able to convert the incident cylindrical beam and form the required irradiance distribution. In this work, the technique of external free-form surface computation is discussed and the optical elements producing uniform irradiance distribution in the rectangular regions (angular sizes are 140° × 67° and 140° × 34°) are designed. The light efficiency of the designed optical elements is larger than 83%.

## 2. Statement of the problem

Assume that the point (compact) light source with intensity function 𝐼(𝛾) is located in the origin of the coordinates. Here *γ* is the zenith angle of the spherical coordinate system. It is required to design an optical element placed above the light source on the condition that the prescribed irradiance distribution 𝐸0(𝑢,𝑣) is produced in the plane 𝑧=𝑓, where 𝐮=(𝑢,𝑣) are the Cartesian coordinates in the target plane.

In order to generate the required irradiance distribution in elongated regions (e.g. a rectangle with large aspect ratio) we propose to use an optical element design shown in Fig. 1 . The surfaces *a*, *b*, and *c* are produced by revolving the collimating profile in Fig. 2 around the *y* axis. Curve *a* of the profile in Fig. 2 is a hyperbola fragment and serves as a collimator, curve *b*produces an imaginary light source at point *M*, and curve *c* is a part of the parabola with the focus at the *M* point; it collimates rays using the total internal reflection effect. The surfaces *a*, *b*, and *c* transform the original spherical beam from the light source to the cylindrical beam with axis *y*. The computation of these surfaces has been detailed in [19].

The external surface *d* in Fig. 1 is assumed to be free form. It should convert the incident cylindrical beam and generate irradiance distribution 𝐸(𝑢,𝑣) maximally close to the required irradiance distribution 𝐸0(𝑢,𝑣).

The external surface can be represented by a parameterized function 𝑟(𝜑,𝑦;𝐩) that defines the distance between the point

𝐫(𝜑,𝑦;𝐩)=(𝑟(𝜑,𝑦;𝐩)sin𝜑; 𝑦; 𝑟(𝜑,𝑦;𝐩)cos𝜑)

of the external surface*d*and the

*y*axis. Here,

*φ*is an angle between the 𝑂𝑦𝑧 plane and the plane containing the surface point r and the

*y*axis. The term “parameterized” means that the shape of the external surface depends on the set of free parameters forming the vector p. Consequently, the irradiance distribution 𝐸(𝑢,𝑣;𝐩) generated in the target plane is defined by the surface parameter p. We consider the design of the external surface

*d*as a minimization problem of the merit function

𝑓(𝐩)=‖𝐸(𝑢,𝑣;𝐩)−𝐸0(𝑢,𝑣)‖→min,

representing the difference between generated and required irradiance distributions. This problem can be solved with use of any well-known gradient search algorithms.## 3. Computation of the irradiance distribution in the target plane

Solving the minimization problem in Eq. (1) requires the solution of the direct problem. The direct problem consists of computation of the irradiance distribution in the target plane 𝑧=𝑓produced by the prescribed optical element with external surface 𝐫(𝜑,𝑦;𝐩).

The profiles of the cylindrical surfaces *a*, *b*, and *c* can be computed analytically [19]. Thus, we assume that the irradiance distribution on the cylindrical wave front generated by these surfaces is known. Consider the imaginary cylindrical surface with radius *R* and the axis *y*corresponding to the wavefront. Let 𝐸′(𝜑,𝑦) be the irradiance distribution on this surface. According to the light flux conservation law, the light flux 𝑑𝛷 that passes through an elementary area 𝑑𝑆 on the imaginary surface equals the light flux reaching the corresponding elementary area 𝑑𝑢𝑑𝑣 in the target plane. Considering the Fresnel loss, the irradiance distribution can be written as

𝑇(𝜑,𝑦;𝐩)𝐸′(𝜑,𝑦)𝑅𝑑𝜑𝑑𝑦=𝐸(𝐮(𝜑,𝑦;𝐩))𝑑𝑢𝑑𝑣,𝐸(𝐮(𝜑,𝑦;𝐩))=𝑇(𝜑,𝑦;𝐩)𝐸′(𝜑,𝑦)𝑅|𝐽(𝐮(𝜑,𝑦;𝐩))|,

where 𝐮(𝜑,𝑦;𝐩) is a target plane point at which the ray passing through the cylindrical surface point (𝜑,𝑦) comes. 𝐽(𝐮(𝜑,𝑦;𝐩))=∂𝑢∂𝜑∂𝑣∂𝑦−∂𝑢∂𝑦∂𝑣∂𝜑 is the Jacobean of transition from the coordinates (𝜑,𝑦) to the (𝑢,𝑣) coordinates. 𝑇(𝜑,𝑦;𝐩) is the Fresnel transmission coefficient.Consider computation of the ray-correspondence functions 𝑢(𝜑,𝑦) and 𝑣(𝜑,𝑦). We introduce a unit vector

𝐚𝟎(𝜑)={sin𝜑,0,cos𝜑},

which defines the direction from the cylindrical wavefront to surface point 𝐫(𝜑,𝑦;𝐩). According to Snell's law, the unit vector 𝐚1 of the refracted ray can be represented as a superposition of the incident ray unit vector 𝐚0 and the normal vector to the surface unit vector n:𝐚𝟏(𝜑,𝑦;𝐩)=𝑛1𝑛2𝐚𝟎(𝜑)+(1−[𝑛1𝑛2𝐚𝟎,𝐧]2−−−−−−−−−−−−−√−𝑛1𝑛2(𝐚𝟎,𝐧))𝐧(𝜑,𝑦;𝐩).

Using Eq. (3) it is easy to deduce the dependencies of the refracted-ray Cartesian coordinates (𝑢,𝑣) on the emitted-ray spherical coordinates (𝜑,𝑦):

𝑢(𝜑,𝑦;𝐩)=𝑟(𝜑,𝑦;𝐩)sin𝜑+𝑎1𝑥(𝜑,𝑦;𝐩)𝑙(𝜑,𝑦;𝐩),𝑣(𝜑,𝑦;𝐩)=𝑦+𝑎1𝑦(𝜑,𝑦;𝐩)𝑙(𝜑,𝑦;𝐩),

where 𝑙(𝜑,𝑦;𝐩)=(𝑓−𝑟(𝜑,𝑦;𝐩)cos𝜑)/𝑎1𝑧(𝜑,𝑦;𝐩) is the distance passed by the refracted ray from the refractive surface to the target plane 𝑧=𝑓.Equation (2) is inconvenient for calculation because it has a singularity when the Jacobian 𝐽(𝐮(𝜑,𝑦;𝐩)) is equal to zero (e.g., in a caustic material). Moreover, Eq. (2) does not allow us to compute the irradiance in the arbitrary point u since the inverse functions 𝜑(𝑢,𝑣) and 𝑦(𝑢,𝑣)are unknown. With use of the following property of the two-dimensional Dirac delta function [20]

𝑓(𝑥,𝑦)|𝐽(𝐮(𝑥,𝑦))||𝑥=𝑥̃𝑦=𝑦̃=∫−∞+∞∫−∞+∞𝑓(𝑥,𝑦)𝛿(𝑢̃−𝑢(𝑥,𝑦),𝑣̃−𝑣(𝑥,𝑦))𝑑𝑥𝑑𝑦,

where (𝑥̃,𝑦̃) is the only point in space (𝑥,𝑦) and 𝑢(𝑥̃,𝑦̃)=𝑢̃ and 𝑣(𝑥̃,𝑦̃)=𝑣̃, the irradiance distribution 𝐸(𝑢,𝑣;𝐩) Eq. (2) can be represented in the integral form [18] as𝐸(𝑢,𝑣;𝐩)=𝑅∫−𝑦max𝑦max∫−𝜋/2𝜋/2𝑇(𝜑,𝑦;𝐩)𝐸′(𝜑,𝑦)𝛿(𝐮−𝐮(𝜑,𝑦;𝐩))𝑑𝜑𝑑𝑦.

For the purpose of numerical computation, the Dirac delta function in Eq. (5) should be replaced by the Gaussian function

𝛿𝜎(𝑢,𝑣)=12𝜋𝜎2exp(−𝑢2+𝑣22𝜎2).

In this case the approximate formula for the irradiance distribution in Eq. (5) can be written as

𝐸(𝑢,𝑣;𝐩)=∫−𝑦max𝑦max∫−𝜋/2𝜋/2𝑅𝑇(𝜑,𝑦;𝐩)𝐸′(𝜑,𝑦)𝛿𝜎(𝐮−𝐮(𝜑,𝑦;𝐩))𝑑𝜑𝑑𝑦.

Equation (6) represents the averaged irradiance distribution in the target plane, and the radius of averaging is defined by the parameter *σ* of the Gaussian function. As distinct from Eq. (2), Eq. (6) has no singularities and allows us to compute the irradiance at any point u.

## 4. Design of optical elements

The technique for calculating the shape of the optical element external surface considered above was implemented in Matlab® R2010b software. The refractive surface 𝑟(𝜑,𝑦;𝐩) was represented as a bicubic spline [21]. For such representation the parameters vector p contains values of the radius vector function *r* and its derivatives ∂𝑟/∂𝜑, ∂𝑟/∂𝑦, and ∂2𝑟/∂𝜑∂𝑦 in the spline nodes. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) gradient method [22] was used for parameter vector optimization. As a merit function, we chose rms deviation of the produced irradiance distribution from the required distribution. The expression for partial derivatives of (1) in this case can be written as

∂𝑓∂𝑝𝑖=1𝑓(𝐩)∬𝑢,𝑣(𝐸(𝑢,𝑣;𝐩)−𝐸0(𝑢,𝑣))∂𝐸(𝑢,𝑣;𝐩)∂𝑝𝑖𝑑𝑢𝑑𝑣,

where∂𝐸(𝑢,𝑣;𝐩)∂𝑝𝑖=∫−𝑦max𝑦max∫−𝜋/2𝜋/2𝑅𝐸′(𝜑,𝑦)∂∂𝑝𝑖(𝑇(𝜑,𝑦;𝐩)𝛿𝜎(𝐮−𝐮(𝜑,𝑦;𝐩)))𝑑𝜑𝑑𝑦.

Detailed derivation of expressions in Eqs. (7) and (8) is presented in [18].

Figure 3 shows the optical element designed to produce uniform irradiance distribution in a 17 × 4 m rectangle located 3 m from the point of the Lambertian light source. Such an optical device can be used for illumination of the elongated hall or corridor. The angle size of the illuminated area is 140° × 67°. The size of the optical element is 58 ×21 × 20 mm along the *x*, *y*, and *z* axes, respectively. The external surface is represented by a bicubic spline with 32 patches. The time of external surface optimization is about 24 minutes for the Intel® Core 2 Quad Q9400 processor.

Fig. 3

Optical element producing the uniform irradiance distribution in a 17 × 4 m rectangle area (overall dimensions are given in millimeters).

Irradiance distribution produced by the designed optical element from Cree® XLamp® XP-G light-emitting diode is shown in Fig. 4 . It was simulated using the illumination design software TracePro®. Hereafter, the irradiance distributions are computed for the following parameters: number of rays used in ray tracing is 100 000, and resolution of the target region is 128 × 128 points. The light efficiency (the part of the emitted light flux that reaches the target plane) of the designed optical element is 83%. The rms deviation of the produced irradiance distribution from the uniform distribution is less than 9%. It should be noted that the optical element was designed for the point light source. The results of the simulation demonstrate the high performance of the element for the Cree® XLamp® extended source. This fact can be explained by the large dimensions of the optical element in comparison with the emitting chip size. It is necessary to emphasize that the TracePro® software is not able to solve the considered optimization problem, and in this paper it is used only for evaluating the performance of the designed optical elements.

Fig. 4

Irradiance distribution in the target plane produced by the optical element in Fig. 3. (a) The grayscale distribution. (b) The irradiance profiles–solid line: 𝑣=0; dashed line, 𝑢=0.

For comparison, we consider an optical element with a single refractive surface that generates uniform irradiance distribution with the same parameters as in the previous case. Such an optical element designed by the method in [18] is shown in Fig. 5 . The size of the optical element is 55 × 30 × 18 mm along the *x*, *y*, and *z* axes, respectively.

Fig. 5

Optical element with a single refractive surface producing the uniform irradiance distribution in a 17 × 4 m rectangle area (overall dimensions are given in millimeters).

Figure 6 depicts the irradiance distribution produced by this element and the same Cree® XLamp® XP-G LED. Because of the limited possibilities of the refractive surface in ray rotation and a small transverse angular size of the illuminated area (~67°), the light efficiency of the optical element in Fig. 5 is smaller than 60%. The rms deviation of the resulting irradiance distribution is 6.7%. A comparison of Fig. 4 and Fig. 6 shows that when generating the irradiance distribution in a rectangular region with a large aspect ratio and a small transverse angular size, the method in [18] fails to provide the light efficiency as high as that of the method proposed in this paper.

. 6

Irradiance distribution in the target plane produced by the optical element in Fig. 5. (a) The grayscale distribution. (b) The irradiance profiles–solid line, 𝑣=0; dashed line, 𝑢=0.

In practice, there is often a requirement to produce irradiance distribution in a shifted rectangular region. Examples could be found in the design of wall-mounted lighting devices for corridors or long halls, street lamps located along the roadside, and so forth. Figure 7 depicts an optical element producing uniform irradiance distribution in a shifted 17 × 2 m rectangle located at a distance of 3 m from the Cree® XLamp® XP-G light LED. The main difference from the previous case is that the center of the illuminated region is not located on the *z* axis but is shifted for a 1 m along the *y* axis from the origin of coordinates. The optical element size is 57 × 21 × 21 mm along the *x*, *y*, and *z* axes, respectively. The time of external surface optimization is less than one and a half hours (the surface is represented by a bicubic spline with 32 patches). Figure 8 shows irradiance distribution produced in the target plane. The light efficiency of the optical element is more than 85%, and the rms deviation of generated irradiance distribution is 7.9%.

Optical element producing the uniform irradiance distribution in a shifted 17 × 2 m rectangle area (overall dimensions are given in millimeters).

Fig. 8

Irradiance distribution in the target plane produced by the optical element in Fig. 7. (a) The grayscale distribution. (b) The irradiance profiles–solid line. 𝑣=0; dashed line, 𝑢=0.

It should be noted that optical elements with a single refractive working surface (Fig. 5) are unable to provide light efficiency higher than 50% while generating shifted irradiance distribution similar to that shown in Fig. 8. In this case, the use of the optical element design proposed in this paper makes possible to increase the light efficiency by 30–40%.

The above examples show that the proposed design and method of external surface computation present a powerful tool for solving problems of illuminating elongated regions with transverse angular sizes up to 70°.

## 5. Conclusion

We developed a method for the design of LED optics intended for producing the required irradiance distribution in elongated rectangular regions with a large aspect ratio. For such regions, the proposed design proves to be more efficient than optical elements with single free-form refractive surfaces. Two optical elements that produce uniformly illuminated regions with sizes 17 × 4 m and 17 × 2 m at a distance of 3 m from the LED light source were designed. The angular sizes of the illuminated regions are 140° × 67° and 140 ° × 34°, respectively. The light efficiency of the computed optical elements exceeds 83%, whereas the rms deviation of the generated irradiance distributions is less than 9%. For such sizes of illuminated regions, the proposed optical element design ensures an increase in light efficiency up to 20–30% compared with the optical elements with single refractive surfaces.

## Acknowledgements

This work was financially supported by the RF Presidential Grant NSh-7414.2010.9, the Russian–American Program “Basic Research and Higher Education” CRDF PG08-014-1, and the RFBR Grant 10-07-00553a.

## References and links

1. G. Pengfei and W. Xu-Jia, “On a Monge–Ampere equation arising in geometric optics,” J. Differential Geom. 48, 205–223 (1998).

2. I. Knowles and Y. Satio, “Radially symmetric solutions of a Monge–Ampere equation arising in the reflector mapping problem,” in *Proceedings of the UAB International Conference on Differential Equations and Mathematical Physics, Lecture Notes in Math* (Springer-Verlag, 1987), pp. 973–1000.

3. V. Oliker and A. Treibergs, *Geometry and Nonlinear Partial Differential Equations* (AMS Bookstore, 1992).

4. W. B. Elmer, “Optical activities in industry,” Appl. Opt. 17, 977–979 (1978). [CrossRef]

5. W. B. Elmer, *The Optical Design of Reflectors* (Wiley, 1980).

6. O. Kusch, *Computer-aided Optical Design of Illumination and Irradiating Devices* (ASLAN Publishing House, 1993).

7. R. A. Hicks, “Designing a mirror to realize a given projection,” J. Opt. Soc. Am. A 22(2), 323–330 (2005). [CrossRef]

8. J. Lautanen, M. Honkanen, V. Kettunen, M. Kuittinen, P. Laakkonen, and J. Turunen, “Wide-angle far-field line generation with diffractive optics,” Opt. Commun. 176(4-6), 273–280 (2000). [CrossRef]

9. L. L. Doskolovich and S. I. Kharitonov, “Calculating the surface shape of mirrors for shaping an image in the form of a line,” J. Opt. Technol. 72(4), 318–321 (2005). [CrossRef]

10. L. L. Doskolovich, N. L. Kazanskiy, and S. Bernard, “Designing a mirror to form a line-shaped directivity diagram,” J. Mod. Opt. 54(4), 589–597 (2007). [CrossRef]

11. J. Bortz, N. Shatz, and D. Pitou, “Optimal design of a nonimaging projection lens for use with an LED source and a rectangular target,” Proc. SPIE 4092, 130–138 (2000). [CrossRef]

12. H. Ries and J. A. Muschaweck, “Tailoring freeform lenses for illumination,” Proc. SPIE 4442, 43–50 (2001). [CrossRef]

13. H. Ries and J. A. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef]

14. B. A. Jacobson and R. D. Gendelbach, “Lens for uniform LED illumination: an example of automated optimization using Monte Carlo ray-tracing of an LED source,” Proc. SPIE 4446, 130–138 (2001).

15. B. Parkyn and D. Pelka, “Free-form illumination lens designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE 6338, 633808, 633808-7 (2006). [CrossRef]

16. A. A. Belousov, L. L. Doskolovich, and S. I. Kharitonov, “A gradient method of designing optical elements for forming a specified irradiance on a curved surface,” J. Opt. Technol. 75(3), 161–165 (2008). [CrossRef]

17. Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008). [CrossRef] [PubMed]

18. M. A. Moiseev and L. L. Doskolovich, “Design of refractive spline surface for generating required irradiance distribution with large angular dimension,” J. Mod. Opt. 57(7), 536–544 (2010). [CrossRef]

19. L. L. Doskolovich and M. A. Moiseev, “Calculations for refracting optical elements for forming directional patterns in the form of a rectangle,” J. Opt. Technol. 76(7), 430–434 (2009). [CrossRef]

20. P. R. Kanwal, *Generalized Functions: Theory and Applications* (Birkhäuser, 2004).

21. C. De Boor, *A Practical Guide to Splines* (Springer, 2001).

22. J. F. Bonnans, *Numerical Optimization: Theoretical and Practical Aspects* (Springer, 2006).