## High resolution irradiance tailoring using multiple freeform surfaces |

Optics Express, Vol. 21, Issue 9, pp. 10563-10571 (2013)

http://dx.doi.org/10.1364/OE.21.010563

Acrobat PDF (980 KB)

### Abstract

More and more lighting applications require the design of dedicated optics to achieve a given radiant intensity or irradiance distribution. Freeform optics has the advantage of providing such a functionality with a compact design. It was previously demonstrated in [Bäuerle et al., Opt. Exp. 20, 14477–14485 (2012)] that the up-front computation of the light path through the optical system (ray mapping) provides a satisfactory approximation to the problem, and allows the design of multiple freeform surfaces in transmission or in reflection. This article presents one natural extension of this work by introducing an efficient optimization procedure based on the physics of the system. The procedure allows the design of multiple freeform surfaces and can render high resolution irradiance patterns, as demonstrated by several examples, in particular by a lens made of two freeform surfaces projecting a high resolution logo (530 × 160 pixels).

© 2013 OSA

## 1. Fine irradiance tailoring remains a challenging task

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

## 2. Ray mapping computation to efficiently find a starting point

_{0}perpendicular to the optical axis via a suitable projection. The target irradiance is equally projected in a consistent way onto a target plane Ω

_{1}. The task of designing a (freeform) optical system can then be described as calculating a diffeomorphism (“ray mapping”) so that the transformed irradiance distribution matches the target distribution: where (

*t*,

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

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

*x*,

*y*) in Ω

_{0}. Figure 1 recalls this principle and shows how a point from the source plane Ω

_{0}is uniquely mapped to a point on the target plane Ω

_{1}.

*ũ*between these two planes can easily be computed using successive linear integrations along the Cartesian coordinate axes [2

2. W. A. Parkyn, “Design of illumination lenses via extrinsic differential geometry,” Proc. SPIE **3428**, 154–162 (1998) [CrossRef] .

**N**to the integrability condition required to build a smooth optical element [3

3. 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] .

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

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

*t*converging towards a stationary irrotational solution (

*t*→ ∞): 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 Poisson’s equation Δ

*f*= −div

*u*

^{⊥}with Dirichlet boundary conditions.

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

*brute-force*ray-tracing). It was demonstrated that this design procedure performs well in cases where homogeneous irradiance of a square area is required, and that the procedure is equally well suited for more complex cases found for example in the automotive industry. The design of a two-sided fog light lens demonstrates the latter [6

6. A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, fresnel-type freeform optics,” Proc. SPIE **8485**, 84850H (2012) [CrossRef] .

## 3. High-resolution tailoring of the irradiance distribution

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

7. A. Bäuerle, R. Wester, C. Schnitzler, and P. Loosen, “Design of efficient freeform lenses for mass-market illumination applications using hybrid algorithms,” Proc. SPIE **7788**, 778807 (2010) [CrossRef] .

*can be written as follows:*

_{S}- for a point source: where
*d*^{2}Φis the elementary light flux,_{S}*d*^{2}Ω the elementary solid angle surrounding the direction of emission (*θ*,*ϕ*), and*I*(*θ*,*ϕ*) is the radiant intensity (in W/sr) of the point source. The elementary solid angle*d*^{2}Ω is approximated by the solid angle encompassed by the three source rays around a triangular face (see Fig. 2). This solid angle remains constant throughout the construction. - for a collimated source: where
*B*(_{S}*x*,*y*) is the source (radiant) irradiance (in W/m^{2}). In this case, the source rays are all directed towards the same direction (e.g. along the*z*-axis) and the cross section of a triangular tube remains constant between the source and the first optical surface.

*can be written as a function of the irradiance (in W/m*

_{T}^{2}) on the target. A triangular flux tube is projected as a triangular area on the target as illustrated in Fig. 2. The flux in this tube can be computed as: where

*A*is the area of the target triangle corresponding to the

_{j}*j*-th face of the surface, and

*B*(

_{T}*x*,

*y*) the desired irradiance at the target point (

*x*,

*y*). This integral is approximated with the simplest quadrature: with (

*x*,

_{j}*y*) being the center of the triangular face on the target as illustrated in Fig. 2.

_{j}### 3.1. Objective function components

### 3.2. Numerical considerations

- the flux difference for a face is multiplied by the number of faces. Since the source flux is already normalized to unity, this ensures the scalability with regards to the number of triangles in the mesh. The effective light flux going through a face hence always has the order of magnitude unity;
- the smoothing component is scaled by the typical size of a triangle area on the target which varies as the square root of the total number of triangles;
- the distance to the target boundary is scaled by the target size.

*h*. It can be shown [8] that on a finite-precision machine with machine precision

*ε*such a computation leads to the following final relative error: with

_{m}*f̃*′(

*x*) the numerical approximation of the derivative and

*ε*the machine round-off precision. The last two terms are competing one with another and dictate an optimum value for

_{m}*h*. Noting

*ε*= 10

_{m}*,*

^{m}*h*= 10

*, 2*

^{n}*f*/(

*hf*′) = 10

*and*

^{u}*f*″(

*x*)/(2

*f*′(

*x*)) = 10

*the optimum value of*

^{v}*n*satisfies

*v*+

*n*= −

*n*+

*u*+

*m*, hence: Typically, with the scaling considerations mentioned above (

*u*,

*v*,

*m*) = (−1, −1, −15) giving a step size

*h*≈ 10

^{−7}and an overall relative computation precision around

*ε*= 10

^{v+n}≈ 10

^{−8}. However, the computation above discards the error induced by the evaluation of

*f*itself (propagation error): a slightly bigger step size (around

*h*≈ 10

^{−6}) gives in practice better results and has been retained in the final implementation.

## 4. Sample applications

### 4.1. Double sided freeform lens

*n*= 1.5) and that a light cone with a half-angle of 45 degrees is captured from the point source.

### 4.2. Logo lens

*mm*

^{2}at a distance of 100

*mm*from the source, and has a resolution of 530 × 160 pixels. We present the results with one and two freeform surfaces, respectively. The optics has a size close to the target one: 50 × 15

*mm*

^{2}.

9. J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE **6670**, 66700A (2007) [CrossRef] .

*I*is the computed irradiance as given by the ray tracing,

_{c}*I*is the prescribed irradiance and

_{p}*I*is a reference irradiance. Bortz and Shatz restricted their computation to an area where the irradiance was sufficiently high and hence took

_{ref}*I*=

_{ref}*I*. In our case, given the contrast at hand (about 4:1) and the fact that the analysis should not be restrained to the letter cores (areas of high irradiance), we computed the RMS with both

_{p}*I*=

_{ref}*I*, the (arithmetical) average of the light irradiance, and

_{avg}*I*=

_{ref}*I*, the peak irradiance found at the core of a letter, for example.

_{max}## 5. Conclusion and future work

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

## Acknowledgments

## References and links

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

2. | W. A. Parkyn, “Design of illumination lenses via extrinsic differential geometry,” Proc. SPIE |

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

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

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

6. | A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, fresnel-type freeform optics,” Proc. SPIE |

7. | A. Bäuerle, R. Wester, C. Schnitzler, and P. Loosen, “Design of efficient freeform lenses for mass-market illumination applications using hybrid algorithms,” Proc. SPIE |

8. | D. Bindel and J. Goodman, “Principles of scientific error computing,” tech. rep., New York University - Computer Science (2009). |

9. | J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(220.2740) Optical design and fabrication : Geometric optical design

(080.1753) Geometric optics : Computation methods

(080.4225) Geometric optics : Nonspherical lens design

(080.4298) Geometric optics : Nonimaging optics

**ToC Category:**

Geometric Optics

**History**

Original Manuscript: March 1, 2013

Revised Manuscript: April 13, 2013

Manuscript Accepted: April 16, 2013

Published: April 23, 2013

**Citation**

Adrien Bruneton, Axel Bäuerle, Rolf Wester, Jochen Stollenwerk, and Peter Loosen, "High resolution irradiance tailoring using multiple freeform surfaces," Opt. Express **21**, 10563-10571 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10563

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### References

- A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express20, 14477–14485 (2012). [CrossRef] [PubMed]
- W. A. Parkyn, “Design of illumination lenses via extrinsic differential geometry,” Proc. SPIE3428, 154–162 (1998). [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]
- H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A19, 590–595 (2002). [CrossRef]
- S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. of Comput. Vision60, 225–240 (2004). [CrossRef]
- A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, fresnel-type freeform optics,” Proc. SPIE8485, 84850H (2012). [CrossRef]
- A. Bäuerle, R. Wester, C. Schnitzler, and P. Loosen, “Design of efficient freeform lenses for mass-market illumination applications using hybrid algorithms,” Proc. SPIE7788, 778807 (2010). [CrossRef]
- D. Bindel and J. Goodman, “Principles of scientific error computing,” tech. rep., New York University - Computer Science (2009).
- J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE6670, 66700A (2007). [CrossRef]

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