## Algorithm for irradiance tailoring using multiple freeform optical surfaces |

Optics Express, Vol. 20, Issue 13, pp. 14477-14485 (2012)

http://dx.doi.org/10.1364/OE.20.014477

Acrobat PDF (1206 KB)

### Abstract

The design of freeform lenses and reflectors allows to achieve non-radially symmetric irradiance distributions whilst keeping the optical system compact. In the case of a point-like source, such as an LED, it is often desired to capture a wide angle of source light in order to increase optical efficiency. This generally results in strongly curved optics, requiring both lens surfaces to contribute to the total ray refraction, and thereby minimising Fresnel losses. In this article, we report on a new design algorithm for multiple freeform optical surfaces based on the theory of optimal mass transport that adresses these requirements and give an example of its application to a problem in general lighting.

© 2012 OSA

## 1. Few published algorithms allow flexible irradiance tailoring

1. J. S. Schruben, “Formulation of a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. **62**, 1498–1501 (1972). [CrossRef]

5. P. Bentez, J. Miano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernndez, J. Alvarez, and W. Falicoff, “SMS design method in 3D geometry: Examples and applications,” Proc. SPIE **5185** (2004). [CrossRef]

6. A. Bruneton, A. Bäuerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE **8167**, 816707 (2011). [CrossRef]

## 2. Ray mapping: relating optical design to transportation theory

*μ*

_{0}on a 2D domain Ω

_{0}(see Fig. 1). Similarly, a target flux density

*μ*

_{1}is given on a domain Ω

_{1}. For the sake of clarity, it shall be assumed that Ω

_{0,1}are parallel 2D planes embedded in ℝ

^{3}and hence

*μ*

_{0}and

*μ*

_{1}are flux densities parametrized by local Cartesian coordinates (

*x,y*) in the respective planes (see Fig. 1).

*t*,

_{x}*t*) represents the target point in Ω

_{y}_{1}to be reached by a source ray passing by (

*x*,

*y*) in Ω

_{0}. Since the light flux along an infinitesimal light tube from the source to the target is conserved, the irradiance transformation can be written: where D

*u*is the Jacobian of

*u*, representing the compression or dilatation of the tube cross-section along the path, and ○ is the usual composition operator. Integrating this equation on the whole domain Ω

_{0}leads to the global energy conservation relation (∫

_{Ω0}

*μ*

_{0}= ∫

_{Ω1}

*μ*

_{1}).

*u*is not unique [7

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

**N**recalled notably by Fournier [8

8. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express **18**, 5295–5304 (2010). [CrossRef] [PubMed]

## 3. Approximating the optimum ray mapping

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

_{0}, thus obtaining a flat flux density

*μ*

_{0}. A modified stereographic projection is typically used to achieve this, at the same time controlling how much of the source light cone should be captured. The target flux distribution is equally projected onto a parallel square domain Ω

_{1}. This is depicted on Fig. 1.

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

*μ*

_{0}and

*μ*

_{1}(typically by two successive 1D numerical integrations along the Cartesian coordinate axes). This results in a starting mapping denoted

*ũ*.

*μ*

_{0}in

*μ*

_{1}can be parametrized with a continuous variable

*t*. Equivalently,

*u*can be regarded as a function of

*t*with

*u*|

_{t}_{=0}=

*ũ*, and its derivative with respect to this variable (i.e. the mapping’s flow) can be computed. The flow obeys the following evolution equation: where D

*u*denotes the mapping’s Jacobian, (

*x*,

*y*)

^{⊥}= (−

*y*,

*x*) represents a rotation by 90 degrees in ℝ

^{2}and Δ

^{−1}div

*u*

^{⊥}denotes the solution

*f*of the Poisson’s equation Δ

*f*= −div

*u*

^{⊥}.

*t*→ ∞) is proven by Haker to be the optimal mapping in the sense of the quadratic cost function and, most importantly, to be irrotationnal. In this article, the equation was solved using the same numerical techniques as employed by Haker and using

*ũ*as a starting point. Note that the evolution equation itself doesn’t make use of

*μ*

_{1}as the corresponding information is already contained in the initial mapping

*ũ*.

## 4. Reconstructing the optical surface from a ray mapping

*i*= 1,...,

*N*whose positions are given by:

**r**

_{0}(

*i*) is the origin of ray

*i*. This can be the position of a point source or the position of the ray after passing through another optical surface.

**s**(

*i*) is the unit direction vector of the ray and

*λ*(

*i*) is a scalar parameter defining the surface point

*i*.

**s**(

*i*) and

**N**(

*i*) give the direction vector

**s**

*(*

_{o}*i*) of the ray after refraction (or reflection) at the surface. The normal vectors

**N**(

*i*) at the triangular surface vertex positions are computed as weighted averages of the normals of the vertex’s neighbor faces.

**s**

*(*

_{o}*i*) and the ray’s position on the optical surface

**r**(

*i*), the point

**T**(

*i*) where the ray intersects the target surface can be computed. The objective function to minimize is thus given by where

*T*(

_{x}*i*) and

*T*(

_{y}*i*) are the actual local target coordinates of the ray for a given vector of parameters

**, and**

*λ**t*(

_{x}*i*) and

*t*(

_{y}*i*), respectively, are the desired local target coordinates as computed by the mapping algorithm.

## 5. Sample application from architectural lighting: wallwashing

### 5.1. On-axis example

*n*= 1.49), minimizing objective function (6). To demonstrate the improvement, the initial mapping was optimized as explained above, and a second set of surfaces was subsequently constructed. Figure 3 shows the corresponding mappings, represented as the deformation of a regular grid, and their respective local curl

*z*components. The overall curl magnitude has been reduced significantly by a factor of about 200.

11. “FRED Software - Optical Engineering,” http://www.photonengr.com.

*z*.

### 5.2. Off-axis configuration

^{2}) as well as the desired homogeneity of the irradiance distribution on the target are respected rather well. In the square area centered on the target and covering 2 m × 2 m, the intensity variation is below 10% on any cross section along the

*y*axis, and below 30% on any cross section along the

*x*axis. The imbalance between these two values arises from the inclination of the optical element with respect to the target plane. Finally, the Monte Carlo simulation shows that over 78% of the source light power is transfered to the target, including all Fresnel losses, thus illustrating the high optical efficiency of the resulting lens. A similar simulation with only one freeform surface leads to greater Fresnel losses (overall optical efficiency reduced to 70%), and fails to acceptably achieve the target distribution as the source light cone is too wide for all the rays to be correctly deflected towards the target.

## 6. Comparing Fresnel losses for designs with one and two freeform surfaces

## 7. Conclusion

## Acknowledgments

## References and links

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

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

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

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

5. | P. Bentez, J. Miano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernndez, J. Alvarez, and W. Falicoff, “SMS design method in 3D geometry: Examples and applications,” Proc. SPIE |

6. | A. Bruneton, A. Bäuerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE |

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

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

9. | L. V. Kantorovich, “On a problem of Monge,” Uspekhi Mat. Nauk. |

10. | L.-C. Evans, “Partial differential equations and Monge-Kantorovich mass transfer,” tech. rep., Department of Mathematics, University of California, Berkeley (2001). |

11. | “FRED Software - Optical Engineering,” http://www.photonengr.com. |

12. | W. Born and E. Wolf, “Basic properties of the electromagnetic field,” in |

**OCIS Codes**

(080.1753) Geometric optics : Computation methods

(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: March 29, 2012

Revised Manuscript: May 8, 2012

Manuscript Accepted: May 31, 2012

Published: June 13, 2012

**Citation**

Axel Bäuerle, Adrien Bruneton, Rolf Wester, Jochen Stollenwerk, and Peter Loosen, "Algorithm for irradiance tailoring using multiple freeform optical surfaces," Opt. Express **20**, 14477-14485 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14477

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### References

- J. S. Schruben, “Formulation of a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am.62, 1498–1501 (1972). [CrossRef]
- H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A19, 590–595 (2002). [CrossRef]
- W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE3482, 389–396 (1998). [CrossRef]
- V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Rational Mech. Anal.201, 1013–1045 (2011). [CrossRef]
- P. Bentez, J. Miano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernndez, J. Alvarez, and W. Falicoff, “SMS design method in 3D geometry: Examples and applications,” Proc. SPIE5185 (2004). [CrossRef]
- A. Bruneton, A. Bäuerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE8167, 816707 (2011). [CrossRef]
- S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis.60, 225–240 (2004). [CrossRef]
- F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express18, 5295–5304 (2010). [CrossRef] [PubMed]
- L. V. Kantorovich, “On a problem of Monge,” Uspekhi Mat. Nauk.3, 225–226 (1948).
- L.-C. Evans, “Partial differential equations and Monge-Kantorovich mass transfer,” tech. rep., Department of Mathematics, University of California, Berkeley (2001).
- “FRED Software - Optical Engineering,” http://www.photonengr.com .
- W. Born and E. Wolf, “Basic properties of the electromagnetic field,” in Principles of Optics7th ed. (Cambridge University Press, 1999), pp. 41–42

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