## Design of compact and smooth free-form optical system with uniform illuminance for LED source

Optics Express, Vol. 18, Issue 9, pp. 9055-9063 (2010)

http://dx.doi.org/10.1364/OE.18.009055

Acrobat PDF (927 KB)

### Abstract

A feedback modification method based on variable separation mapping is proposed in the design of free-form optical system with uniform illuminance for LED source. In this method, the non-negligible size of LED source is taken into account, and a smooth optical system is established with single freeform surface regenerated by adding feedback to the lens design for a point light source. More rounds of feedback can improve the lens performance. As an example, a smooth free-form lens with rectangular illuminance distribution is designed, and the illuminance uniformity is improved from 18.75% to 81.08% after eight times feedback.

© 2010 OSA

## 1. Introduction

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

9. L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. **46**(18), 3716–3723 (2007). [CrossRef] [PubMed]

9. L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. **46**(18), 3716–3723 (2007). [CrossRef] [PubMed]

10. P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. **43**, 1489–1502 (2004). [CrossRef]

9. L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. **46**(18), 3716–3723 (2007). [CrossRef] [PubMed]

*E’*(

_{1}*x,y*) is obtained through simulation with the actual (extended) LED source. Let

*E*

_{0}(

*x,y*) be the desired illuminance distribution, then by employing a Feedback modification function

*η*(

_{i}*x,y*)=

*η*(

_{i}*E’*(

_{i}*x,y*),

*E’*(

_{i-1}*x,y*),…,

*E’*(

_{1}*x,y*),

*E*(

_{0}*x,y*)) wherein i = 1, the free-form surface is regenerated for the new illuminance distribution

*η*·E

_{1}_{0}(

*x,y*). System performance improves with increasing i, the number of iterations. In this method, there is no need to introduce discontinuities to control normal errors. In fact, deviations of illuminance distribution caused by the size of LED source and surface construction can be virtually eliminated after several feedbacks. As an example, a smooth free-form lens with rectangular illuminance distribution is designed, and the illuminance uniformity is improved from 18.75% to 81.08% after eight feedbacks.

## 2. Feedback modification method

### 2.1 Energy conservation

*I*(

*u,v*) and

*E*(

*x,y*) denote the intensity distribution of the light source in the direction (

*u,v*) and the desired illuminance distribution in the target plane, respectively. Thus, energy conservation of a lossless optical system can be expressed as Eq. (1) [9

**46**(18), 3716–3723 (2007). [CrossRef] [PubMed]

*u,v*)| and |J (

*x,y*)| are the absolute Jacobian factors.

*x,y*) in the target plane can be obtained from the polar coordinates (

*u,v*) of the free-form surface expressed as Eq. (2) and Eq. (3):

*u,v*) from (

*x,y*) expressed as Eq. (4) and Eq. (5), reversely [11

11. Y. Han, X. Zhang, Z. Feng, K. Qian, H. Li, Y. Luo, X. Li, G. Huang, and B. Zhu, “Variable-separation three dimensional freeform nonimaging optical system design based on target-to-source mapping and micro belt surface construction, ” Sciencepaper Online 1–9(2010). http://www.paper.edu.cn/en/paper.php?serial_number=201002-443

### 2.2 Freeform surface construction

**T**(

*x,y,z*), wherein z is constant; the freeform surface is also separated into the same number of cells, and the polar coordinates (

*u,v*) of these cells’ points

**P**(

*r,u,v*) can be calculated from Eq. (4) and Eq. (5); Then the freeform surface construction is reduced to solve the lengths

*r*of these points to the origin point O.

**P**(

*r*) on curve

_{0},u_{0},v_{0}**C**(

*j*) is a known quantity, and it’s corresponding point on the target plane is

**T**(

*x*); Taking

_{0},y_{0},z_{0}**In**(

*u*) as the unit incident vector, the refractive vector

_{0},v_{0}**Out**and normal vector

**N**at point

**P**(

*r*) can be calculated from Eq. (6) and Eq. (7): Wherein n is the refractive index of the freeform lens.

_{0},u_{0},v_{0}*r*of point

**P**(

*r,u,v*) on the next curve

**C**(

*j+1*) can be derived from Eq. (8):

*1*), then all the points and curves can be obtained from Eqs. (4)~(8); The initial curve is constructed as follows: assuming an initial point, the next point on curve C(

*1*) is then calculated as the intersection of the light ray with the tangent plane of the previous point [9

**46**(18), 3716–3723 (2007). [CrossRef] [PubMed]

10. P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. **43**, 1489–1502 (2004). [CrossRef]

**46**(18), 3716–3723 (2007). [CrossRef] [PubMed]

### 2.3 Feedback modification

*x,y*) and polar coordinates (

*u,v*) with the desired illuminance distribution

*E*(

_{0}*x,y*). Then the optical model is calculated in 2.2, and simulation with the LED source using the optical model is performed, the simulation result

*E’*(

*x,y*) may have a large deviation from

*E*(

_{0}*x,y*) because of the extended light source and the normal errors. The aim of this letter is to design the optical model with a new given illuminance distribution

*E*(

*x,y*), which can make the simulation result approach the desired one,

*E*(

_{0}*x,y*). An effective way named Feedback modification method is introduced to obtain

*E*(

*x,y*). The idea of this method can be described as follows: during the (

*i+1)-th*iteration, the given illuminance distribution

*E*(

_{0}*x,y*) is modified through the feedback function

*η*(

_{i}*x,y*)

*= η*(

_{i}*E’*(

_{i}*x,y*),

*E’*(

_{i-1}*x,y*)

*,…, E’*(

_{1}*x,y*),

*E*(

_{0}*x,y*)), wherein

*E’*(

_{i}*x,y*) is the simulation result of the

*i-th*iteration. The relationship between the new given illuminance distribution

*E*(

_{i + 1}*x,y*) for

*(i+1)-th*iteration and

*E*(

_{0}*x,y*) can be written as Eq. (9):

*E’*(

_{i}*x, y*)=0,

*E’*(

_{i}*x,y*) <<

*E*(

_{0}*x,y*) or

*E’*(

_{i}*x,y*) >>

*E*(

_{0}*x,y*) at somewhere on the target plane, there will be a dramatic change in the value of

*η*(

_{i}*x,y*). To avoid this from happening, a feedback function with lower and upper tolerance based on Eq. (10) is introduced as Eq. (11) and Eq. (12): wherein

*r*and

_{1}*r*are the lower and upper tolerance, respectively.

_{2}*E*), we can get new correspondence between the rectangular coordinates (

_{i+1}(x,y*x,y*) and polar coordinates (

*u,v*); New freeform lens model can be regenerated based on this new correspondence.

## 3. Design example

*E*(

_{0}*l,m*),

*l*=1,2,3,…,40,

*m*=1,2,3,…,20. The feedback function shown in Eq. (11) and Eq. (12) can also be written as Eq. (13) and Eq. (14): wherein the lower tolerance

*r*=0.5, the upper tolerance

_{1}*r*=2, and

_{2}*l*=1,2,3,…,40,

*m*=1,2,3,…,20. The lower and upper tolerance values are decided by the actual simulation process, and those values that result in the simulation results converging to the desired distribution with the least feedback times should be selected. In this design example, better simulation results could be obtained by setting lower and upper tolerance value as 0.5 and 2, respectively.

*i*)

*-th*simulation result

*E’*(

_{i}*l,m*) is produced based on Monte Carlo method, and the illuminance uniformity is defined as Eq. (15):wherein

*E’*and

_{imin}*E’*are the minimum and average illuminance value of the matrix

_{iaverage}*E’*(

_{i}*l,m*), l=1,2,3,…,40, m = 1,2,3,…,20.

*xz*plane and

*yz*plane are shown in Fig. 8c and 8d, which clearly show that the dimension of the final lens is broadening in

*xz*direction and

*yz*direction while the central height maintains 7 mm.

## 4. Conclusions

## Acknowledgments

## References and links

1. | R. Winston, J. C. Miñano, and P. Benítez, eds., with contributions by N. Shatz and J. C. Bortz, eds., |

2. | J. S. Schruben, “Formulation a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. A |

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

4. | A. Timinger, J. Muschaweck, and H. Ries, “Designing tailored free-form surfaces for general illumination,” Proc. SPIE |

5. | V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed illuminance properties,” Proc. SPIE |

6. | L. Caffarelli, S. Kochengin, and V. Oliker, “On the numerical solution of the problem of reflector design with given far-field scattering data,” Contemp. Math. |

7. | X.-J. Wang, “On design of a reflector antenna II,” Calculus Var. Partial Differ. Equ. |

8. | W. A. Parkyn, “The design of illumination lenses via extrinsic differential geometry,” Proc. SPIE |

9. | L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. |

10. | P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. |

11. | Y. Han, X. Zhang, Z. Feng, K. Qian, H. Li, Y. Luo, X. Li, G. Huang, and B. Zhu, “Variable-separation three dimensional freeform nonimaging optical system design based on target-to-source mapping and micro belt surface construction, ” Sciencepaper Online 1–9(2010). http://www.paper.edu.cn/en/paper.php?serial_number=201002-443 |

12. | L. Piegl, and W. Tiller, The NURBS Book 2nd, ed (Springer-Verlag, Berlin, 1997). |

**OCIS Codes**

(220.2945) Optical design and fabrication : Illumination design

(080.4225) Geometric optics : Nonspherical lens design

(080.4298) Geometric optics : Nonimaging optics

**History**

Original Manuscript: March 2, 2010

Revised Manuscript: April 6, 2010

Manuscript Accepted: April 6, 2010

Published: April 15, 2010

**Citation**

Yi Luo, Zexin Feng, Yanjun Han, and Hongtao Li, "Design of compact and smooth free-form optical system with uniform illuminance for LED source," Opt. Express **18**, 9055-9063 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-9055

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### References

- R. Winston, J. C. Miñano, and P. Benítez, eds., with contributions by N. Shatz and J. C. Bortz, eds., Nonimaging Optics (Elsevier, 2005), Chapter 7.
- J. S. Schruben, “Formulation a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. A 6, 1498–1501 (1972).
- H. Ries and J. A. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef]
- A. Timinger, J. Muschaweck, and H. Ries, “Designing tailored free-form surfaces for general illumination,” Proc. SPIE 5186, 128–132 (2003). [CrossRef]
- V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed illuminance properties,” Proc. SPIE 5942, 594207 (2005). [CrossRef]
- L. Caffarelli, S. Kochengin, and V. Oliker, “On the numerical solution of the problem of reflector design with given far-field scattering data,” Contemp. Math. 226, 13–32 (1999).
- X.-J. Wang, “On design of a reflector antenna II,” Calculus Var. Partial Differ. Equ. 20(3), 329–341 (2004). [CrossRef]
- W. A. Parkyn, “The design of illumination lenses via extrinsic differential geometry,” Proc. SPIE 3482, 191–193 (1998).
- L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef] [PubMed]
- P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1502 (2004). [CrossRef]
- Y. Han, X. Zhang, Z. Feng, K. Qian, H. Li, Y. Luo, X. Li, G. Huang, and B. Zhu, “Variable-separation three dimensional freeform nonimaging optical system design based on target-to-source mapping and micro belt surface construction, ” Sciencepaper Online 1–9(2010). http://www.paper.edu.cn/en/paper.php?serial_number=201002-443
- L. Piegl and W. Tiller, The NURBS Book 2nd, ed (Springer-Verlag, Berlin, 1997).

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