## Freeform LED lens for uniform illumination

Optics Express, Vol. 16, Issue 17, pp. 12958-12966 (2008)

http://dx.doi.org/10.1364/OE.16.012958

Acrobat PDF (368 KB)

### Abstract

Light flux from LED must be redistributed to meet the needs of lighting in most cases, a new method is proposed for its secondary optic design. Based on refractive equation and energy conservation, a set of first-order partial differential equations which represent the characters of LED source and desired illumination were presented. The freeform lens was constructed by solving these equations numerically. The numerical results showed that we can get a freeform lens for the illumination of uniformity near to 90%, with considerable high computation speed. This method can shorten the designing time of the freeform lens with high accepted tolerance.

© 2008 Optical Society of America

## 1. Introduction

3. 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]

5. B. Parkyn and D. Pelka, “Free-form illumination lens designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE **6338**, 633808 (2005). [CrossRef]

## 2. Partial differential equation sets

### 2.1 Refractive vectors

*,*

**N***and*

**I***as the unit vectors of normal, incident and refractive vectors respectively, and they can be expressed as:*

**O****has connections with the position of point p and t. After using Snell’s law at the point p, we have the equations as shown in Eqs. (4) and (5):**

*O*_{I}and n

_{O}indicate the refractive index of the incidence and emergence medium.

**can be expressed by ρ, θ and φ. Equations (4) and (5) are the partial differential equations used for the freeform lens calculation. And the subscript of**

*I***include the first-order partial derivative of ρ(θ, φ) on the directions of θ and φ, which stands for the slope at point p of the freeform lens [11]. It could be calculated from:**

*N**p*is the vector component of

**on x, y and z directions. In orthogonal coordinates they can be expressed as:**

*p*### 2.2 Energy conservation

*φ*:

_{MAX}*E*(

**) is the luminance at point t, A is the area illuminated.**

*t**I*(

**(**

*I**φ*)) is LED emitting intensity in the direction of

**(**

*I**φ*). Equation (8) indicates the relationship between θ, φ and x, y, z, and its exact form depends on the topological mapping from the source to the target plane.

**T**is fixed, θ and φ can be gotten though Eq. (8). After replacing the corresponding items in Eqs. (4) and (5) by θ and φ, the first-order partial differential equation sets of ρ(θ, φ) on the directions of θ and φ can be deduced. It is difficult to obtain the analytic solution from these partial differential equations, so numerical methods are employed [12].

## 3. Freeform lens design

^{2}[13

13. Lumileds LED technical data sheet, “Luxeon star technical data sheet” (Lumileds, 2006). http://www.lumileds.com/pdfs/DS23.pdf.

**T**is a rectangle with the ratio of 4:3. It’s perpendicular to the z-axis, and its center is (0, 0, 30).

_{S}, and angels of incidence and refractive on the spherical surface are α

_{1}and α

_{2}, just as Fig. 2(b) shows. It’s easy to get that:

_{S}, α

_{1}, α

_{2}can be shown as:

_{1}, α

_{2}can be expressed by φ

_{S}through analytic geometry and the refraction law. If a point p on freeform surface is known, for example, the vertex point, then φ

_{S}can be got according to Eq. (11). Thereby get

**through Eqs. (9) and (10).**

*I**φ*directions, and can be considered as constant in the range of

*φ*

_{n}±d

*φ*/2. Assume the intensity in direction of

*φ*

_{n}as

*I*(

**(**

*I**φ*

_{n})), and the number of measuring points is N, then the luminous flux of LED between

*φ*

_{0}and

*φ*

_{N}can be expressed as:

*φ*=(

*φ*

_{N}-

*φ*

_{0})/N,

*φ*

_{n}=

*φ*

_{0}+n*d

*φ*. In uniform illumination,

*E*(

**) in Eq. (8) is a constant. The topological mapping applied is showed in Fig. 3, which means lights of equal φ would be refracted to the same edge of the rectangle [14**

*t*14. W. A. Parkyn, “Segmented illumination lenses for steplighting and wall-washing,” Proc. SPIE **3779**, 363–370 (1999). [CrossRef]

_{S}is confirmed, Φ

_{LED}would be known. Then θ in the first quadrant can be expressed as:

_{S}through Eq. (13) from X, Y, dθ and dφ are applied to discretize the source, Then the polar and azimuth angle of light from source become:

**became function of ρ(θ, φ), so that the incident position on the target plane can be calculated through Eqs. (12), (13) and (14). Thereby the unknowns of Eqs. (4) and (5) become ρ(θ, φ) and its first-order derivatives in the direction of θ, φ. Using the difference scheme of ρ(θ, φ) to replace its first derivatives, while discretizing the equations with φ**

*I*_{S}and θ

_{S}and the grids depicted in Fig. 3, the partial differential equation sets turn into a set of nonlinear equations whose unknown is ρ(θ

_{S}, φ

_{S}).

15. H. Chase, “Optical Design with Rotationally Symmetric NURBS,” Proc. SPIE **4832**, 10–24 (2002). [CrossRef]

16. T. L. R. Davenport, “3D NURBS representation of surface for illumination,” Proc. SPIE **4832**, 293–301 (2002). [CrossRef]

## 4. Tolerance analysis

^{2}Lambertian rectangle, one can get the target illumination as shown in the Fig. 7. Comparing with the result in Fig. 5, the uniformity has been influenced when the radiation turn to Lambertian. Figure 7(b) shows the illumination uniformity across the center is about 80%, and uniformity on the quarter line near the diagonal region is 75%. In fact the illumination difference between MM1D and a Lambertian source is much larger than the fluctuation among the real sources, so the uniformity would be higher than 80% in actual application.

## 5. Conclusion

## References and links

1. | P. Zhou, W. Lu, Y. X. Lin, Z. R. Zheng, H. F. Li, and P. F. Gu, “Fly eye lens array used in liquid crystal projection display with high light efficiency,” Acta Optica Sinica |

2. | M. Shen, H. F. Li, W. Lu, and X. Liu, “Method of reflective fly eye lens design for LED illuminating projection system,” Acta Photonica Sinica |

3. | 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 |

4. | B. A. Jacobson and R. D. Gengelbach, “Lens for uniform LED illumination: an example of automated optimization using Monte Carlo ray-tracing of an LED source,” Proc. SPIE |

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

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

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

8. | Y. Ding and P. F. Gu, “The Freeform Reflector for Uniform Illumination,” Acta Optica Sinica |

9. | Y. Ding, X. Liu, H. F. Li, and P. F. Gu, “The design of the freeform reflector for uniform illumination,” in |

10. | J. Schruben, “Formulation of a reflector design problem for a lighting fixture,” J. Opt. Soc. Am |

11. | W. H. Chen, Introduction of Differential Geometry (Beijing University, 1990), Chap. 4. |

12. | Y. C. Su and Q. G. Wu, Numerical Solutions of Partial Differential Equations (Weather, 1989), Chap. 1. |

13. | Lumileds LED technical data sheet, “Luxeon star technical data sheet” (Lumileds, 2006). http://www.lumileds.com/pdfs/DS23.pdf. |

14. | W. A. Parkyn, “Segmented illumination lenses for steplighting and wall-washing,” Proc. SPIE |

15. | H. Chase, “Optical Design with Rotationally Symmetric NURBS,” Proc. SPIE |

16. | T. L. R. Davenport, “3D NURBS representation of surface for illumination,” Proc. SPIE |

17. | T. L. R. Davenport, “Generation of NC Tool Path for Subdivision Surface,” in |

18. | Precitech product features, “Freeform 700G” (Precitech, 2006). http://www.precitech.com/Precitech_ff700G_features.html. |

19. | Y. Z. Wang and L. J. Chen, “A real-time NURBS surface interpolator for 5-axis surface machining,” Chinese Journal of Aeronautics |

**OCIS Codes**

(220.2945) Optical design and fabrication : Illumination design

(080.4225) Geometric optics : Nonspherical lens design

(080.4298) Geometric optics : Nonimaging optics

**ToC Category:**

Geometric optics

**History**

Original Manuscript: May 20, 2008

Revised Manuscript: July 6, 2008

Manuscript Accepted: July 21, 2008

Published: August 11, 2008

**Citation**

Yi Ding, Xu Liu, Zhen-rong Zheng, and Pei-fu Gu, "Freeform LED lens for uniform illumination," Opt. Express **16**, 12958-12966 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12958

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### References

- P. Zhou, W. Lu, Y. X. Lin, Z. R. Zheng, H. F. Li, and P. F. Gu, "Fly eye lens array used in liquid crystal projection display with high light efficiency," Acta Optica Sinica 24, 587-591 (2004).
- M. Shen, H. F. Li, W. Lu, and X. Liu, "Method of reflective fly eye lens design for LED illuminating projection system," Acta Photonica Sinica 35,93-95 (2006).
- 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]
- B. A. Jacobson and R. D. Gengelbach, "Lens for uniform LED illumination: an example of automated optimization using Monte Carlo ray-tracing of an LED source," Proc. SPIE 4446, 130-138 (2002).
- B. Parkyn and D. Pelka, "Free-form illumination lens designed by a pseudo-rectangular lawnmower algorithm," Proc. SPIE 6338, 633808 (2005). [CrossRef]
- H. Ries and J. Muschaweck, "Tailoring freeform lenses for illumination," Proc. SPIE 6338, 633808 (2001).
- H. Ries and J. Muschaweck, "Tailored freeform optical surfaces," J. Opt. Soc. Am. A 19, 590-595 (2002). [CrossRef]
- Y. Ding and P. F. Gu, "The Freeform Reflector for Uniform Illumination," Acta Optica Sinica 27, 540-544 (2007).
- Y. Ding, X. Liu, H. F. Li, and P. F. Gu, "The design of the freeform reflector for uniform illumination," in Proceedings of Asia Display 2007, Volume 1. (Shanghai, China, 2007), pp. 735-738.
- J. Schruben, "Formulation of a reflector design problem for a lighting fixture," J. Opt. Soc. Am 62, 1498-1501 (1972). [CrossRef]
- W. H. Chen, Introduction of Differential Geometry (Beijing University, 1990), Chap. 4.
- Y. C. Su and Q. G. Wu, Numerical Solutions of Partial Differential Equations (Weather, 1989), Chap. 1.
- Lumileds LED technical data sheet, "Luxeon star technical data sheet" (Lumileds, 2006). http://www.lumileds.com/pdfs/DS23.pdf.
- W. A. Parkyn, "Segmented illumination lenses for steplighting and wall-washing," Proc. SPIE 3779, 363-370 (1999). [CrossRef]
- H. Chase, "Optical Design with Rotationally Symmetric NURBS," Proc. SPIE 4832, 10-24 (2002). [CrossRef]
- T. L. R. Davenport, "3D NURBS representation of surface for illumination," Proc. SPIE 4832, 293-301 (2002). [CrossRef]
- T. L. R. Davenport, " Generation of NC Tool Path for Subdivision Surface," in Proceedings of CAD/Graphics 2001, Q. Peng, ed. (International Academic, Kunming, China, 2001), pp. 1-7.
- Precitech product features, "Freeform 700G" (Precitech, 2006). http://www.precitech.com/Precitech_ff700G_features.html.
- Y. Z. Wang and L. J. Chen, "A real-time NURBS surface interpolator for 5-axis surface machining," Chinese Journal of Aeronautics 18, 263-272 (2005). [CrossRef]

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