## Designing double freeform optical surfaces for controlling both irradiance and wavefront |

Optics Express, Vol. 21, Issue 23, pp. 28693-28701 (2013)

http://dx.doi.org/10.1364/OE.21.028693

Acrobat PDF (1403 KB)

### Abstract

We propose an improved double freeform-optical-surface design method for shaping a prescribed irradiance distribution whilst forming a desired wavefront from a given incident beam. This method generalizes our previous work [Opt. Exp. 21, 14728-14735 (2013)] to tackle non-separable beam irradiances. We firstly compute a proper ray mapping using an adaptive mesh method in the framework of the L^{2} Monge-Kantorovich mass transfer problem. Then, we construct the two freeform optical surfaces according to this mapping using a modified simultaneous point-by-point procedure which is aimed to minimize the surface errors. For the first surface, the modified procedure works by firstly approximating a value to the next point by only using the slope of the current point and then improving it by utilizing both slopes of the two points based on Snell’s law. Its corresponding point on the second surface can be computed using the constant optical path length condition. A design example of producing a challenging irradiance distribution and a non-ideal wavefront demonstrates the effectiveness of the method.

© 2013 Optical Society of America

## 1. Introduction

1. W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE **3482**, 389–396 (1998). [CrossRef]

13. R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. **38**(2), 229–231 (2013). [CrossRef] [PubMed]

19. Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express **21**(12), 14728–14735 (2013). [CrossRef] [PubMed]

7. 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]

*et al*. for powerfully coupling two input wavefronts into two output ones [20

20. 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**(7), 1489–1502 (2004). [CrossRef]

^{2}Monge-Kantorovich problem. Then, we develop a modified simultaneous point-by-point procedure to construct the two freeform surfaces with smaller errors. In section 3, we verify this method by showing a challenging design example of projecting a complex irradiance distribution and a non-ideal wavefront. Finally, a brief summary is given in section 4.

## 2. Design method

*z*direction. The input beam at

*z*= 0 is supposed to have an irradiance distribution

*I*(

_{in}*x*,

*y*) and the wavefront close to it is denoted by position vectors

**S**= (

*x*,

*y*,

*z*). The output beam irradiance at

*z*=

*d*is prescribed by

*I*(

_{out}*x'*,

*y'*) and the wavefront close to it is denoted by

**T**= (

*x'*,

*y'*,

*z'*).

*n*,

_{1}*n*and

_{0}*n*are set as the the refractive indices of three mediums divided by the two freeform surfaces. The design goal is to specify

_{2}**P**= (

*x*,

_{1}*y*,

_{1}*z*) and

_{1}**Q**= (

*x*,

_{2}*y*,

_{2}*z*) on the two freeform surfaces in order to realize the above transformations. The two steps of the design are shown in 2.1 and 2.2, respectively.

_{2}### 2.1 Ray mapping computation

*x' = f*(

*x*) as

*y'*=

*g*(

*y*) [7

7. 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]

*x'*=

*f*(

*x, y*) and

*y'*=

*g*(

*x*,

*y*). By introducing this ray mapping into the right side of Eq. (1), we can obtain Eq. (2):wherein

*J*(

*f*,

*g*) is the Jacobian matrix of the ray mapping. Equation (2) alone can’t uniquely specify the ray mapping and need to be intertwined with the geometry of the optical elements for a rigorous solution [14

14. H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE **5876**, 587607 (2005). [CrossRef]

18. V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy **3**(1), 035599 (2013). [CrossRef]

*L*Monge-Kantorovich problem can help to specify an appropriate ray mapping (see e.g. [21

^{2}21. J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math. **84**(3), 375–393 (2000). [CrossRef]

23. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. **61**(3), 298–307 (2011). [CrossRef]

22. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. **60**(3), 225–240 (2004). [CrossRef]

*et al.*for approximating an optimum ray mapping in their irradiance tailoring approach [12

12. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express **20**(13), 14477–14485 (2012). [CrossRef] [PubMed]

*et al.*recently [23

23. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. **61**(3), 298–307 (2011). [CrossRef]

*L*Monge-Kantorovich problem can be characterized as the gradient of a convex potential

^{2}*u*(

*x*,

*y*) as shown in Eq. (3) [24

24. R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pac. J. Math. **17**(3), 497–510 (1966). [CrossRef]

25. R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J. **80**(2), 309–323 (1995). [CrossRef]

23. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. **61**(3), 298–307 (2011). [CrossRef]

*u*(t→∞) solves Eq. (4). Thus, the ray mapping can be determined by taking the spatial gradient of

_{∞}*u*.

_{∞}**S**and the input ray vectors

**In,**and the output ray sequences can be defined by the output wavefront position vectors

**T**and the output ray vectors

**Out**[19

19. Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express **21**(12), 14728–14735 (2013). [CrossRef] [PubMed]

### 2.2 Double freeform-surface construction

19. Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express **21**(12), 14728–14735 (2013). [CrossRef] [PubMed]

*y*-direction (or

*x*-direction) and then along the

*x*-direction (or

*y*-direction) by intersecting the next input ray with the tangent plane of the current point while the second surface is obtained by equaling the optical path lengths (OPLs) between the input and output wavefronts. The construction of the first freeform surface is very analogous to Euler’s method for solving ODEs. The difference is that it is a two-dimensional integration of a partial differential equation (PDEs) using the language of extrinsic differential geometry [1

1. W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE **3482**, 389–396 (1998). [CrossRef]

**21**(12), 14728–14735 (2013). [CrossRef] [PubMed]

**P**

_{i,1}and

**Q**

_{i,1}of the two initial curves, wherein

*i*= 1,2,3,…or n-1. We can describe the procedure by computing the next points

**P**

_{i + 1,1}and

**Q**

_{i + 1,1}. We can firstly calculate the normal vector

**N**

_{i,1}based on Snell’s law so that the ray emitted from

**P**

_{i,1}can be redirected to

**Q**

_{i,1}. Next, we compute

**P**

*'*

_{i + 1,1}, an approximate value of

**P**

_{i + 1,1}, by intersecting the input ray vector

**In**

_{i + 1,1}with the tangent plane of

**P**

_{i,1}. Once we have

**P**

*'*

_{i + 1,1}, we can get it’s corresponding point

**Q**

*'*

_{i + 1,1}on the second surface immediately based on the constancy of the OPLs between the input and output wavefronts. Next, we can acquire the normal vector

**N**

*'*

_{i + 1,1}at

**P**

*'*

_{i + 1,1}so that the ray emitted from

**P**

*'*

_{i + 1,1}can reach

**Q**

*'*

_{i + 1,1}. Next, we need to compute an intermediate point

**P**

_{i + 1/2,1}by intersecting

**In**

_{i + 1/2,1}with the tangent plane of

**P**

_{i,1}, wherein

**In**

_{i + 1/2,1}is emitted from the midpoint of

**S**

_{i,1}and

**S**

_{i + 1,1}. After that, we can obtain

**P**

_{i + 1,1}by intersecting

**In**

_{i + 1,1}with the plane through

**P**

_{i + 1/2,1}and perpendicular to

**N**

*'*

_{i + 1,1}. Then, we can get

**Q**

_{i + 1,1}using the constant OPL condition. We can generate the entire two initial curves by repeat the process described above.

**P**

_{i,2}and

**Q**

_{i,2}of the next two curves from the two initial curves, wherein

*i*= 1,2,3,…n. We firstly calculate

**P**

*'*

_{i,2}by intersecting

**In**

_{i,2}with the tangent plane of

**P**

_{i,1}. Next, we can get

**Q**

*'*

_{i,2}based on the constancy of the OPLs. After that, We can acquire

**N**

*'*

_{i,2}based on Snell’s law in vector form. Then, we obtain

**P**

_{i,2}from the knowledge of

**N**

_{i,1},

**N**

*'*

_{i,2}and an intermediate point

**P**

_{i,3/2}. We repeat this process to generate all the curves and interpolate them into the entire two freeform surfaces. Figure 3 illustrates the flow diagram of this point-by-point procedure.

## 3. Design example

*λ*, wherein the beam wavelength

*λ*= 1064nm (see Fig. 5(b)). The input laser beam is supposed to have a Gaussian irradiance distribution (beam waist: 5mm) and a spherical wavefront with a PV of 100λ.

*x*=

*f'*(

*x',y'*) and

*y*=

*g'*(

*x',y'*) since the output irradiance distribution is much more complicated than the input one. The final adaptive mesh of the input irradiance is shown in Fig. 6(a). A uniform Cartesian mesh of the output irradiance is also presented in Fig. 6(b) for better visualizing the Energy transfer. No surprisingly, the ray mapping is more strongly deformed around the regions corresponding to the edges of the letters where the target irradiance is essentially discontinuous.

28. M. Gdeisat and F. Lilley, “Two-dimensional phase unwrapping problem,” http://www.ljmu.ac.uk/GERI/CEORG_Docs/Two_Dimensional_Phase_Unwrapping_Final.pdf.

*I*is quantified by the relative root-mean-square deviation (RRMSD) from the target irradiance, as shown in Eq. (6):The resulting output wavefront can be assessed by the residual root-mean-square (RMS) error shown in Eq. (7):wherein

_{S}*z'*is the

_{s}*z*-coordinate value of the simulated output wavefront.

## 4. Conclusion

**21**(12), 14728–14735 (2013). [CrossRef] [PubMed]

**61**(3), 298–307 (2011). [CrossRef]

^{2}Monge-Kantorovich framework. Then, we integrate the two freeform optical surfaces using a modified simultaneous point-by-point procedure to reduce the surface errors. Simulations of a challenging example yielded promising results in terms of the desired irradiance and wavefront distributions.

## Acknowledgments

## References and links

1. | W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE |

2. | W. A. Parkyn, and Jr., “Illuminating lens designed by extrinsic differential geometry,” Teledyne Lighting and Display Products, Inc., US Patent 5924788 (1999). |

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

4. | W. A. Parkyn and D. G. Pelka, “Free-form lenses for rectangular illumination zones,” Anthony, Inc., US Patent 7674019 (2010). |

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

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

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

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

9. | Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express |

10. | F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express |

11. | D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. |

12. | A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express |

13. | R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. |

14. | H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE |

15. | T. Glimm and V. Oliker, “Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat’s principle,” Indiana Univ. Math. J. |

16. | V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal. |

17. | J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A |

18. | V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy |

19. | Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express |

20. | 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. |

21. | J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math. |

22. | S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. |

23. | M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. |

24. | R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pac. J. Math. |

25. | R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J. |

26. | W. B. Elmer, |

27. | J. W. Goodman, |

28. | M. Gdeisat and F. Lilley, “Two-dimensional phase unwrapping problem,” http://www.ljmu.ac.uk/GERI/CEORG_Docs/Two_Dimensional_Phase_Unwrapping_Final.pdf. |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(080.1753) Geometric optics : Computation methods

(080.4225) Geometric optics : Nonspherical lens design

(080.4298) Geometric optics : Nonimaging optics

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: September 13, 2013

Revised Manuscript: October 31, 2013

Manuscript Accepted: November 4, 2013

Published: November 14, 2013

**Citation**

Zexin Feng, Lei Huang, Guofan Jin, and Mali Gong, "Designing double freeform optical surfaces for controlling both irradiance and wavefront," Opt. Express **21**, 28693-28701 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28693

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### References

- W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE3482, 389–396 (1998). [CrossRef]
- W. A. Parkyn, and Jr., “Illuminating lens designed by extrinsic differential geometry,” Teledyne Lighting and Display Products, Inc., US Patent 5924788 (1999).
- B. Parkyn and D. Pelka, “Free form lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE6338, 633808 (2006). [CrossRef]
- W. A. Parkyn and D. G. Pelka, “Free-form lenses for rectangular illumination zones,” Anthony, Inc., US Patent 7674019 (2010).
- H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A19(3), 590–595 (2002). [CrossRef] [PubMed]
- V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed illuminance properties,” Proc. SPIE5942, 594207 (2005). [CrossRef]
- 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]
- Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express16(17), 12958–12966 (2008). [CrossRef] [PubMed]
- Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express18(9), 9055–9063 (2010). [CrossRef] [PubMed]
- F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express18(5), 5295–5304 (2010). [CrossRef] [PubMed]
- D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett.36(6), 918–920 (2011). [CrossRef] [PubMed]
- A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express20(13), 14477–14485 (2012). [CrossRef] [PubMed]
- R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett.38(2), 229–231 (2013). [CrossRef] [PubMed]
- H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE5876, 587607 (2005). [CrossRef]
- T. Glimm and V. Oliker, “Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat’s principle,” Indiana Univ. Math. J.53(5), 1255–1277 (2004). [CrossRef]
- V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal.201(3), 1013–1045 (2011). [CrossRef]
- J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A24(2), 463–469 (2007). [CrossRef] [PubMed]
- V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy3(1), 035599 (2013). [CrossRef]
- Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express21(12), 14728–14735 (2013). [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(7), 1489–1502 (2004). [CrossRef]
- J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math.84(3), 375–393 (2000). [CrossRef]
- S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis.60(3), 225–240 (2004). [CrossRef]
- M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math.61(3), 298–307 (2011). [CrossRef]
- R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pac. J. Math.17(3), 497–510 (1966). [CrossRef]
- R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J.80(2), 309–323 (1995). [CrossRef]
- W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, 1980), Chap.4.
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).
- M. Gdeisat and F. Lilley, “Two-dimensional phase unwrapping problem,” http://www.ljmu.ac.uk/GERI/CEORG_Docs/Two_Dimensional_Phase_Unwrapping_Final.pdf .

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