## Design of high-efficient freeform LED lens for illumination of elongated rectangular regions |

Optics Express, Vol. 19, Issue S3, pp. A225-A233 (2011)

http://dx.doi.org/10.1364/OE.19.00A225

Acrobat PDF (1328 KB)

### Abstract

We propose a method for the design of an optical element generating the required irradiance distribution in a rectangular area with a large aspect ratio. Application fields include streetlights, the illumination of halls or corridors, and so forth. The design assumes that the optical element has a complex form and contains two refractive surfaces. The first one converts a spherical beam from the light source to a cylindrical beam. The second one transforms an incident cylindrical beam and generates the required irradiance distribution in the target plane. Two optical elements producing a uniform irradiance distribution from a Cree® XLamp® source in rectangular regions of 17 m × 4 m and 17 m × 2 m are designed. The light efficiency of the designed optical element is larger than 83%, whereas the irradiance nonuniformity is less than 9%.

© 2011 OSA

## 1. Introduction

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

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]

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]

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]

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]

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]

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]

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]

**76**(7), 430–434 (2009). [CrossRef]

**76**(7), 430–434 (2009). [CrossRef]

## 2. Statement of the problem

*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

*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

**76**(7), 430–434 (2009). [CrossRef]

*d*in Fig. 1 is assumed to be free form. It should convert the incident cylindrical beam and generate irradiance distribution

*d*and the

*y*axis. Here,

*φ*is an angle between the

**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

**p**. We consider the design of the external surface

*d*as a minimization problem of the merit functionrepresenting 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

*a*,

*b*, and

*c*can be computed analytically [19

**76**(7), 430–434 (2009). [CrossRef]

*R*and the axis

*y*corresponding to the wavefront. Let

**n**:

**u**since the inverse functions

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]

*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

**p**contains values of the radius vector function

*r*and its derivatives

**57**(7), 536–544 (2010). [CrossRef]

*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.

**57**(7), 536–544 (2010). [CrossRef]

*x*,

*y*, and

*z*axes, respectively.

**57**(7), 536–544 (2010). [CrossRef]

*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%.

## 5. Conclusion

## Acknowledgements

## References and links

1. | G. Pengfei and W. Xu-Jia, “On a Monge–Ampere equation arising in geometric optics,” J. Differential Geom. |

2. | I. Knowles and Y. Satio, “Radially symmetric solutions of a Monge–Ampere equation arising in the reflector mapping problem,” in |

3. | V. Oliker and A. Treibergs, |

4. | W. B. Elmer, “Optical activities in industry,” Appl. Opt. |

5. | W. B. Elmer, |

6. | O. Kusch, |

7. | R. A. Hicks, “Designing a mirror to realize a given projection,” J. Opt. Soc. Am. A |

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. |

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. |

10. | L. L. Doskolovich, N. L. Kazanskiy, and S. Bernard, “Designing a mirror to form a line-shaped directivity diagram,” J. Mod. Opt. |

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 |

12. | H. Ries and J. A. Muschaweck, “Tailoring freeform lenses for illumination,” Proc. SPIE |

13. | H. Ries and J. A. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A |

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 |

15. | B. Parkyn and D. Pelka, “Free-form illumination lens designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE |

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. |

17. | Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express |

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. |

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. |

20. | P. R. Kanwal, |

21. | C. De Boor, |

22. | J. F. Bonnans, |

**OCIS Codes**

(080.2740) Geometric optics : Geometric optical design

(080.4298) Geometric optics : Nonimaging optics

**History**

Original Manuscript: February 10, 2011

Revised Manuscript: March 10, 2011

Manuscript Accepted: March 10, 2011

Published: March 23, 2011

**Citation**

Mikhail A. Moiseev, Leonid L. Doskolovich, and Nikolay L. Kazanskiy, "Design of high-efficient freeform LED lens for illumination of elongated rectangular regions," Opt. Express **19**, A225-A233 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-S3-A225

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### References

- G. Pengfei and W. Xu-Jia, “On a Monge–Ampere equation arising in geometric optics,” J. Differential Geom. 48, 205–223 (1998).
- 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.
- V. Oliker and A. Treibergs, Geometry and Nonlinear Partial Differential Equations (AMS Bookstore, 1992).
- W. B. Elmer, “Optical activities in industry,” Appl. Opt. 17, 977–979 (1978). [CrossRef]
- W. B. Elmer, The Optical Design of Reflectors (Wiley, 1980).
- O. Kusch, Computer-aided Optical Design of Illumination and Irradiating Devices (ASLAN Publishing House, 1993).
- R. A. Hicks, “Designing a mirror to realize a given projection,” J. Opt. Soc. Am. A 22(2), 323–330 (2005). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- H. Ries and J. A. Muschaweck, “Tailoring freeform lenses for illumination,” Proc. SPIE 4442, 43–50 (2001). [CrossRef]
- H. Ries and J. A. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef]
- 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).
- B. Parkyn and D. Pelka, “Free-form illumination lens designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE 6338, 633808, 633808-7 (2006). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- P. R. Kanwal, Generalized Functions: Theory and Applications (Birkhäuser, 2004).
- C. De Boor, A Practical Guide to Splines (Springer, 2001).
- J. F. Bonnans, Numerical Optimization: Theoretical and Practical Aspects (Springer, 2006).

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