Abstract

In illumination applications, freeform lens design starts by modeling the light source intensity, then determining the lens surface. Therefore, any change in the source intensity distribution directly impacts the created illuminance. Freeform lens-array could reduce the dependency between the illuminance pattern and the source intensity distribution. It divides the light source energy into separate beams; each creates the same illuminance pattern. All separated beams are superimposed at a distant target to generate the final illuminance pattern. This paper explains the design procedures of a freeform lens-array in detail, simulates the optical performance, and experimentally evaluates the working principle. Experiments evaluate using different light sources and using a variable illuminated aperture. Moreover, simulations compare the freeform lens-array to a traditional freeform lens.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Freeform lenses perform beam-shaping to create asymmetrical illuminance patterns [1]. Optical designers often consider the one-to-one mapping approach in freeform lenses [26]. In this design approach, each incident ray from the light source is mapped to a specific position at a distant target, as shown in Fig. 1.This design approach’s drawback is the sensitivity of changing the created illuminance patterns to the imperfections in the source intensity distribution.

 

Fig. 1. (a) Beam-shaping, considering the one-to-one design approach, and the light energy at a distant target, and (b) Dividing the light source energy into parts, then redistributing and superimposing the light intensity using freeform lens-lets creating a structured illuminance

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Reducing the one-to-one mapping dependency could be achieved by considering the many-to-one design approach. It is often used to create uniform distributions [713]. This approach’s working principle divides the light energy into separate beams using lens-lets, redistributes, and superimposes those beams at the distant target. The light energy superimposition compensates for the errors (e.g., imperfections in the source intensity).

This paper aims at three objectives. First, Combining the advantages of both the one-to-one and the many-to-one design approaches to create a structured asymmetrical illuminance. The combined approach was discussed as a theory only by simulations in [14]. Second, explaining the design procedures. Third, evaluating the optical performance experimentally.

2. Design procedures

The purpose is to design a refractive lens-array (7×7 lens-lets) to redistribute collimated rays. It creates a structured illuminance representing Fig. 2(a) of an area 20×20 mm2 at a distant target of 200 mm, as an example. Design procedures of every single lens-let (4×4 mm2) start by determining the source-target one-to-one mapping.

 

Fig. 2. (a) The image of the Town Musicians of Bremen, (b) Uniform grid representing the light source collimated irradiance, and (c) Adaptive grid representing the required mapping

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2.1 Source-target one-to-one mapping

The principle of energy conservation assumes transferring the total light source energy without any losses through the lens. It can be represented mathematically by

$$\int {I_S}({\boldsymbol \xi } )d{\boldsymbol \xi }\; = \int {I_T}({\boldsymbol x} )d{\boldsymbol x},$$
where IS (ξ) is the light source intensity, IT (x) is the target light intensity, ξ represents the coordinates of the input intensity distribution, and x represents the coordinates of the created output intensity distribution, where
$${\boldsymbol \xi } = {({\xi ,\; \eta \; } )^\textrm{T}},\; \textrm{and}\; {\boldsymbol x} = {({x,y\; } )^\textrm{T}}.\; \; $$

The mapping data are determined using the steady-state solution proposed by Sulman et al. [15] to solve

$$\textrm{det}{\nabla ^2}\varPsi ({\boldsymbol \xi } )\; {I_T}({\nabla \varPsi ({\boldsymbol \xi } )} )= \; {I_S}({\boldsymbol \xi } ),$$
where ($\nabla \varPsi $) is the required mapping to be determined. Equation (2) describes the redistribution of source-target mapping during the beam-shaping process. After introducing the time dependency represented by
$$\frac{{\textrm{d}\varPsi }}{{dt}} = \textrm{log}\left( {\; \frac{{\textrm{det}{\nabla^2}\varPsi ({\boldsymbol \xi } )\; {I_T}({\nabla \varPsi ({\boldsymbol \xi } )} )}}{{{I_S}({\boldsymbol \xi } )}}} \right),$$
the numerical steady-state solution of the required mapping can be determined using the standard centered second-order finite difference scheme and the forward Euler scheme for spatial and temporal discretization, respectively, in an iteration process [15]. To guarantee the convergence of the numerical solution, we normalize ${I_T}({\boldsymbol x} )$. In this way, the conservation of energy condition is fulfilled where
$${I_S}({\boldsymbol \xi } )= 1.$$

After performing the iteration process, the numerical solution of Eq. (3) is determined, generating the adaptive grid shown in Fig. 2(c). The time required for generating this adaptive grid (Grid size: 401×401) is almost 60 minutes using Intel Core i7-7500U CPU @ 2.70GHz processor and a 16.0 GB RAM.

2.2 Surface sag determination

The unit directional vector i = (ix, iy, iz)T shown in Fig. 3 represents the incident ray. Each intersection point in the grid shown in Fig. 2(b) illustrates the incident ray’s position. Figure 2(c) illustrates the required distant target’s mapped position. The surface gradients represented by

$$\nabla {\textbf S}({x,\; y} )= {\left( {\frac{{\partial \textrm{S}}}{{\partial \textrm{x}}},{\boldsymbol \; }\frac{{\partial \textrm{S}}}{{\partial \textrm{y}}}} \right)^\textrm{T}},$$
are determined using the vector form of Snell’s law
$${n_1}\; ({{\textbf i}\; \textrm{x}\; {\textbf n}} )= \; {n_2}{\boldsymbol \; }\; ({{\textbf r}\; \textrm{x}\; {\textbf n}} ),$$
where n1 and n2 are the refractive indices of mediums. Surface gradients are then corrected using an iteration process to fulfill the surface’s integrability condition [2,16]. Finally, integrating the surface gradients determines the surface sag values S(x, y) shown in Fig. 4(b).

 

Fig. 3. Simulation setup

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Fig. 4. (a) Surface gradients, (b) Sag data of one lens-let, and (c) Sag data of the lens-array

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2.3 Freeform lens-array

The first step in determining the total sag values is arranging all freeform lens-lets in an array (28 × 28 mm2). Then, applying a focusing correction by adding the sag values represented by

$${z_f}\; ={-} \left( {\frac{{{x^2} + {y^2}}}{{200}}} \right),$$
to the total lens-array. This leads to the superimposition of the light energy at the required distance. Finally, all points outside the diameter of 28 mm are excluded. The final created illuminance pattern is the result of superimposing all patterns. The boundaries of the individual lens-let are not identical; this leads to surface discontinuity at the freeform lens-array borders, as explained in our paper [17]. This problem is solved by enforcing the integrability condition at the borders by correcting the gradient data according to [16,18].

2.4 Simulation results

A light point source and a Plano-convex lens generate the collimated beam, and a CAD file models the freeform lens-array (PMMA material). A rectangular detector detects the illuminance pattern at a 200 mm distance. Figure 5(c) shows the created illuminance pattern by tracing 1.0E7 rays using OpticStudio [19].

 

Fig. 5. (a) Simulation setup, (b) Input irradiance, and (c) Simulation results

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3. Experiments and analysis

Figure 6(a) shows the fabricated freeform lens-array using a highly dynamic diamond turning process. The experimental setup in Fig. 6(b) consists of a fiber-coupled LED, a Plano-convex lens (SILL A1ADX0620), and a blank camera sensor (Allied Vision Prosilica GT 4907). The boundaries of the created illuminance pattern deviate slightly from the simulation results due to the difficulties of fabricating the sag changes at the lens-lets’ boundaries.

 

Fig. 6. (a) Fabricated freeform lens-array, (b) Experimental setup including the collimated beam, fabricated freeform lens-array, and blank camera sensor, and (c) Experimental results

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3.1 Different light sources

Three light sources from different generations (WU-8-56SEC, LUMEX SSL-LXA228SIC, BROADCOM ASMT-BR20) illuminate the lens-array individually using the same experimental setup. The collimated lights’ experimental irradiances do not match the datasheets, as shown in Fig. 7(a, b). Figure 7(c) shows the detected illuminance pattern using each light source. Figure 7(d) shows the error correlation between the created patterns and the reference pattern Fig. 6(c). Results indicate no direct dependence between the illuminance patterns and the source irradiances because each light source irradiances has concentrated intensity spots different from the others. Nevertheless, intensity spots are not reflecting directly in the error correlation images. Blurring appears due to the light collimation quality (Extended light sources), compared to results shown in Fig. 6(c).

 

Fig. 7. Three different light sources: (a) Irradiance according to the datasheet, (b) Irradiance measured using Goniophotometer, (c) Created illuminance patterns [Camera black-red scale], (d) Error correlation [Jet-Colormap [20]]

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3.2 Variable aperture

This section evaluates the influence of partial illumination on the created patterns. Changing the active aperture area is performed by varying the Iris diameter from 1 mm to 28 mm, then detecting the illuminance pattern corresponding to each aperture, as shown in Fig. 8(b). To evaluate the influence of changing the diameter, Fig. 6(c) compares the detected patterns as a reference pattern. The error correlation represents the difference between each detected pattern and the reference, as shown in Fig. 8 (c). The Root Mean Square (RMS) of the error correlation patterns are determined and represented, as shown in Fig. 8 (d).

$$\textrm{RMS} = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^n a_i^2}}{n}} ,$$
where ${a_i}$ is each value in the error pattern, and n is the number of data points. The RMS values show the area limits of dependence between the active aperture area and the required illuminance pattern.

 

Fig. 8. (a) Schematic diagram (Iris diaphragm), (b) Created illuminance pattern correspond to each diameter, (c) Error Correlation, and (c) Root mean square values (see Visualization 1)

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One freeform lens-let (4×4 mm2) is the minimum required illumination to create the required illuminance pattern. Below this limit, beam-shaping follows the one-to-one design approach. The direct relation of one-to-one mapping appears in the first three patterns, as shown in Fig. 8 (b). By increasing the aperture radius covering more than one freeform lens-let, beam-shaping follows the combined design approach. The last four patterns in Fig. 8(b) show the possibility of creating an illuminance pattern that works almost-independently of the active aperture area. The degree of dependency decreases by increasing the active illuminated area due to the superimposed patterns’ increment.

A staircase pattern (Circular illuminated rings) appears in the third image in Fig. 8(b). This illuminance pattern is created by illuminating the middle freeform lens-let. The first guess was due to lens fabrication errors. Therefore, a new experiment was performed by illuminating each lens-let separately and then detecting the corresponding illuminance pattern. The experimental setup (light source and detector) is explained in Fig. 6(b), but each lens-let is illuminated separately using a square aperture.

Figure 9(a) presents the detected illuminance patterns after arranging them in the same order of freeform lens-lets. The staircase pattern appears clearly after arranging all patterns in the same order of freeform lens-lets, as shown in Fig. 9. Such staircase patterns appear due to fabrication parameters, for example, the material feeding speed and the selected diamond turning tool, as explained in Chapter 7 in [21]. What is interesting for us in this paper, the light superimposition compensates for the fabrication errors.

$${\textrm{Final}\; {\textrm{illuminance}}\; {\textrm{pattern}}} = \mathop \sum \limits_{i = 1}^n {{\boldsymbol p}_{\boldsymbol i}},$$
where ${{\boldsymbol p}_{\boldsymbol i}}{\boldsymbol \; }$ is each illuminance pattern data, and n is the number of patterns.

 

Fig. 9. Superimposition of all illuminance patterns

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3.3 Comparison to the one-to-one design approach

This section investigates, by simulation, the freeform lens-array’s optical performance compared to the traditional one-to-one freeform lens. The simulation setup is as explained in section 2.4. Both lenses have the same diameter, redistribute the light to create illuminance patterns representing Fig. 2(a). Figure 10(c) shows the error correlation patterns’ RMS values, following the same IRIS experiment procedures in section 3.2. The reference patterns in determining the RMS values are the created patterns by illuminating both lenses’ full aperture separately. Therefore, the RMS values in Fig. 10(c) equal to zero in case of illuminating the full aperture. Results visualize the difference between both design approaches. The freeform lens-array can be considered in applications that might expect obstacles between the light source and the secondary optics, or even in applications that might need to vary the active illuminated aperture.

 

Fig. 10. (a) Freeform lens – Sag data and Simulation results, (b) Freeform lens-array – Sag data and Simulation results, and (c) RMS values of error correlation (see Visualization 2).

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4. Conclusion

This paper is an explanation and experimental evaluation of combining two design approaches to create a structured illuminance pattern. The combination aims to reduce the dependence between the created illuminance and the input intensity distributions. The simulation results through design procedures lead to fabricating the freeform lens-array.

Experiments evaluate the influence of using three different light sources. The change in the light source distribution does not directly impact the created patterns as often occurs in the one-to-one freeform traditional lenses. The blurring effect sometimes appears due to the quality of light collimation, not the intensity distribution. Changing the active illuminated aperture shows promising results in applications that may expect obstacles between the light source and the secondary optics, or even in applications that may need to vary the active aperture. The superimposition principle compensates for the errors of the diamond turning marks due to the fabrication process. Finally, an optical simulation compares the freeform lens-array to a traditional freeform lens. Results show that the freeform lens-array reduces the dependency of the created illuminance on the source intensity distributions.

Funding

Bremen University of Applied Sciences; Bundesministerium für Bildung und Forschung.

Disclosures

The authors declare no conflicts of interest.

References

1. J. Chaves, Introduction to nonimaging optics (CRC Press, 2008).

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

3. Y. Ding, X. Liu, Z.-r. Zheng, and P.-f. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008). [CrossRef]  

4. Z. Zhenrong, H. Xiang, and L. Xu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48(35), 6627–6634 (2009). [CrossRef]  

5. J.-J. Chen, Z.-Y. Huang, T.-S. Liu, M.-D. Tsai, and K.-L. Huang, “Freeform lens design for light-emitting diode uniform illumination by using a method of source-target luminous intensity mapping,” Appl. Opt. 54(28), E146–E152 (2015). [CrossRef]  

6. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18(5), 5295–5304 (2010). [CrossRef]  

7. W. J. Cassarly, S. R. David, D. G. Jenkins, A. P. Riser, and T. L. R. Davenport, “Automated design of a uniform distribution using faceted reflectors,” Opt. Eng 39(7), 1830 (2000). [CrossRef]  

8. P. Schreiber, S. Kudaev, P. Dannberg, and U. D. Zeitner, “Homogeneous LED-illumination using microlens arrays,” in Optics & Photonics 2005, R. Winston and R. J. Koshel, eds. (SPIE, 2005), 59420 K.

9. E. Aslanov, L. L. Doskolovich, and M. A. Moiseev, “Thin LED collimator with free-form lens array for illumination applications,” Appl. Opt. 51(30), 7200 (2012). [CrossRef]  

10. L. Wang, J. M. Sasián, P. Su, and R. J. Koshel, “Generation of uniform illumination using faceted reflectors,” in Nonimaging Optics: Efficient Design for Illumination and Solar Concentration VI, R. Winston and J. M. Gordon, eds. (SPIE, 2009), 74230Y.

11. S. R. David, T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” in International Optical Design Conference 1998, L. R. Gardner and K. P. Thompson, eds. (SPIE, 1998), p. 437.

12. J.-W. Pan, C.-M. Wang, H.-C. Lan, W.-S. Sun, and J.-Y. Chang, “Homogenized LED-illumination using microlens arrays for a pocket-sized projector,” Opt. Express 15(17), 10483 (2007). [CrossRef]  

13. W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” in Design of Efficient Illumination Systems, R. J. Koshel, ed. (SPIE, 2003), p. 133.

14. M. Essameldin, T. Binkele, F. Fleischmann, T. Henning, and W. Lang, “Functional concept for the source independent beam-shaping of LED light,” OSA Continuum 2(3), 759 (2019). [CrossRef]  

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

16. A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of the tenth IEEE international conference on computer vision, ICCV 2005 (IEEE, 2005), 174–181 Vol. 1.

17. M. Essameldin, F. Fleischmann, and T. Henning, “Problems of using the PMA adaptive mesh method in lens-array design for LED signal lighting,” in David Kidger, (Ed.), 2018 – Illuminaton Optics V, p. 25.

18. N. Petrovic, I. Cohen, B. J. Frey, R. Koetter, and T. S. Huang, “Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models,” in 2001 IEEE conference on computer vision and pattern recognition (IEEE Comput. Soc, 2001), I-743–I-748.

19. “Zemax-LLC, \Zemax Opticstudio,” http://www.zemax.com/os/opticstudio.

20. J. D. Hunter, “Matplotlib: A 2D Graphics Environment,” Comput. Sci. Eng. 9(3), 90–95 (2007). [CrossRef]  

21. X. Jiang and P. J. Scott, Advanced metrology. Freeform surfaces / X. Jane Jiang and Paul J. Scott, eds., (Academic Press, 2020).

References

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  1. J. Chaves, Introduction to nonimaging optics (CRC Press, 2008).
  2. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002).
    [Crossref]
  3. Y. Ding, X. Liu, Z.-r. Zheng, and P.-f. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
    [Crossref]
  4. Z. Zhenrong, H. Xiang, and L. Xu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48(35), 6627–6634 (2009).
    [Crossref]
  5. J.-J. Chen, Z.-Y. Huang, T.-S. Liu, M.-D. Tsai, and K.-L. Huang, “Freeform lens design for light-emitting diode uniform illumination by using a method of source-target luminous intensity mapping,” Appl. Opt. 54(28), E146–E152 (2015).
    [Crossref]
  6. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18(5), 5295–5304 (2010).
    [Crossref]
  7. W. J. Cassarly, S. R. David, D. G. Jenkins, A. P. Riser, and T. L. R. Davenport, “Automated design of a uniform distribution using faceted reflectors,” Opt. Eng 39(7), 1830 (2000).
    [Crossref]
  8. P. Schreiber, S. Kudaev, P. Dannberg, and U. D. Zeitner, “Homogeneous LED-illumination using microlens arrays,” in Optics & Photonics 2005, R. Winston and R. J. Koshel, eds. (SPIE, 2005), 59420 K.
  9. E. Aslanov, L. L. Doskolovich, and M. A. Moiseev, “Thin LED collimator with free-form lens array for illumination applications,” Appl. Opt. 51(30), 7200 (2012).
    [Crossref]
  10. L. Wang, J. M. Sasián, P. Su, and R. J. Koshel, “Generation of uniform illumination using faceted reflectors,” in Nonimaging Optics: Efficient Design for Illumination and Solar Concentration VI, R. Winston and J. M. Gordon, eds. (SPIE, 2009), 74230Y.
  11. S. R. David, T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” in International Optical Design Conference 1998, L. R. Gardner and K. P. Thompson, eds. (SPIE, 1998), p. 437.
  12. J.-W. Pan, C.-M. Wang, H.-C. Lan, W.-S. Sun, and J.-Y. Chang, “Homogenized LED-illumination using microlens arrays for a pocket-sized projector,” Opt. Express 15(17), 10483 (2007).
    [Crossref]
  13. W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” in Design of Efficient Illumination Systems, R. J. Koshel, ed. (SPIE, 2003), p. 133.
  14. M. Essameldin, T. Binkele, F. Fleischmann, T. Henning, and W. Lang, “Functional concept for the source independent beam-shaping of LED light,” OSA Continuum 2(3), 759 (2019).
    [Crossref]
  15. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of the Monge–Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
    [Crossref]
  16. A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of the tenth IEEE international conference on computer vision, ICCV 2005 (IEEE, 2005), 174–181 Vol. 1.
  17. M. Essameldin, F. Fleischmann, and T. Henning, “Problems of using the PMA adaptive mesh method in lens-array design for LED signal lighting,” in David Kidger, (Ed.), 2018 – Illuminaton Optics V, p. 25.
  18. N. Petrovic, I. Cohen, B. J. Frey, R. Koetter, and T. S. Huang, “Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models,” in 2001 IEEE conference on computer vision and pattern recognition (IEEE Comput. Soc, 2001), I-743–I-748.
  19. “Zemax-LLC, \Zemax Opticstudio,” http://www.zemax.com/os/opticstudio .
  20. J. D. Hunter, “Matplotlib: A 2D Graphics Environment,” Comput. Sci. Eng. 9(3), 90–95 (2007).
    [Crossref]
  21. X. Jiang and P. J. Scott, Advanced metrology. Freeform surfaces / X. Jane Jiang and Paul J. Scott, eds., (Academic Press, 2020).

2019 (1)

2015 (1)

2012 (1)

2011 (1)

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

2010 (1)

2009 (1)

2008 (1)

2007 (2)

2002 (1)

2000 (1)

W. J. Cassarly, S. R. David, D. G. Jenkins, A. P. Riser, and T. L. R. Davenport, “Automated design of a uniform distribution using faceted reflectors,” Opt. Eng 39(7), 1830 (2000).
[Crossref]

Agrawal, A.

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of the tenth IEEE international conference on computer vision, ICCV 2005 (IEEE, 2005), 174–181 Vol. 1.

Anselm, C.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” in Design of Efficient Illumination Systems, R. J. Koshel, ed. (SPIE, 2003), p. 133.

Aslanov, E.

Binkele, T.

Cassarly, W. J.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18(5), 5295–5304 (2010).
[Crossref]

W. J. Cassarly, S. R. David, D. G. Jenkins, A. P. Riser, and T. L. R. Davenport, “Automated design of a uniform distribution using faceted reflectors,” Opt. Eng 39(7), 1830 (2000).
[Crossref]

S. R. David, T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” in International Optical Design Conference 1998, L. R. Gardner and K. P. Thompson, eds. (SPIE, 1998), p. 437.

Chang, J.-Y.

Chaves, J.

J. Chaves, Introduction to nonimaging optics (CRC Press, 2008).

Chellappa, R.

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of the tenth IEEE international conference on computer vision, ICCV 2005 (IEEE, 2005), 174–181 Vol. 1.

Chen, J.-J.

Cohen, I.

N. Petrovic, I. Cohen, B. J. Frey, R. Koetter, and T. S. Huang, “Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models,” in 2001 IEEE conference on computer vision and pattern recognition (IEEE Comput. Soc, 2001), I-743–I-748.

Dannberg, P.

P. Schreiber, S. Kudaev, P. Dannberg, and U. D. Zeitner, “Homogeneous LED-illumination using microlens arrays,” in Optics & Photonics 2005, R. Winston and R. J. Koshel, eds. (SPIE, 2005), 59420 K.

Davenport, T. L. R.

W. J. Cassarly, S. R. David, D. G. Jenkins, A. P. Riser, and T. L. R. Davenport, “Automated design of a uniform distribution using faceted reflectors,” Opt. Eng 39(7), 1830 (2000).
[Crossref]

David, S. R.

W. J. Cassarly, S. R. David, D. G. Jenkins, A. P. Riser, and T. L. R. Davenport, “Automated design of a uniform distribution using faceted reflectors,” Opt. Eng 39(7), 1830 (2000).
[Crossref]

S. R. David, T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” in International Optical Design Conference 1998, L. R. Gardner and K. P. Thompson, eds. (SPIE, 1998), p. 437.

Ding, Y.

Doskolovich, L. L.

Essameldin, M.

M. Essameldin, T. Binkele, F. Fleischmann, T. Henning, and W. Lang, “Functional concept for the source independent beam-shaping of LED light,” OSA Continuum 2(3), 759 (2019).
[Crossref]

M. Essameldin, F. Fleischmann, and T. Henning, “Problems of using the PMA adaptive mesh method in lens-array design for LED signal lighting,” in David Kidger, (Ed.), 2018 – Illuminaton Optics V, p. 25.

Fleischmann, F.

M. Essameldin, T. Binkele, F. Fleischmann, T. Henning, and W. Lang, “Functional concept for the source independent beam-shaping of LED light,” OSA Continuum 2(3), 759 (2019).
[Crossref]

M. Essameldin, F. Fleischmann, and T. Henning, “Problems of using the PMA adaptive mesh method in lens-array design for LED signal lighting,” in David Kidger, (Ed.), 2018 – Illuminaton Optics V, p. 25.

Fournier, F. R.

Frey, B. J.

N. Petrovic, I. Cohen, B. J. Frey, R. Koetter, and T. S. Huang, “Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models,” in 2001 IEEE conference on computer vision and pattern recognition (IEEE Comput. Soc, 2001), I-743–I-748.

Gu, P.-f.

Henning, T.

M. Essameldin, T. Binkele, F. Fleischmann, T. Henning, and W. Lang, “Functional concept for the source independent beam-shaping of LED light,” OSA Continuum 2(3), 759 (2019).
[Crossref]

M. Essameldin, F. Fleischmann, and T. Henning, “Problems of using the PMA adaptive mesh method in lens-array design for LED signal lighting,” in David Kidger, (Ed.), 2018 – Illuminaton Optics V, p. 25.

Huang, K.-L.

Huang, T. S.

N. Petrovic, I. Cohen, B. J. Frey, R. Koetter, and T. S. Huang, “Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models,” in 2001 IEEE conference on computer vision and pattern recognition (IEEE Comput. Soc, 2001), I-743–I-748.

Huang, Z.-Y.

Hunter, J. D.

J. D. Hunter, “Matplotlib: A 2D Graphics Environment,” Comput. Sci. Eng. 9(3), 90–95 (2007).
[Crossref]

Jenkins, D. G.

W. J. Cassarly, S. R. David, D. G. Jenkins, A. P. Riser, and T. L. R. Davenport, “Automated design of a uniform distribution using faceted reflectors,” Opt. Eng 39(7), 1830 (2000).
[Crossref]

Jiang, X.

X. Jiang and P. J. Scott, Advanced metrology. Freeform surfaces / X. Jane Jiang and Paul J. Scott, eds., (Academic Press, 2020).

Knoflach, C.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” in Design of Efficient Illumination Systems, R. J. Koshel, ed. (SPIE, 2003), p. 133.

Koetter, R.

N. Petrovic, I. Cohen, B. J. Frey, R. Koetter, and T. S. Huang, “Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models,” in 2001 IEEE conference on computer vision and pattern recognition (IEEE Comput. Soc, 2001), I-743–I-748.

Koshel, R. J.

L. Wang, J. M. Sasián, P. Su, and R. J. Koshel, “Generation of uniform illumination using faceted reflectors,” in Nonimaging Optics: Efficient Design for Illumination and Solar Concentration VI, R. Winston and J. M. Gordon, eds. (SPIE, 2009), 74230Y.

Kudaev, S.

P. Schreiber, S. Kudaev, P. Dannberg, and U. D. Zeitner, “Homogeneous LED-illumination using microlens arrays,” in Optics & Photonics 2005, R. Winston and R. J. Koshel, eds. (SPIE, 2005), 59420 K.

Lan, H.-C.

Lang, W.

Liu, T.-S.

Liu, X.

Moiseev, M. A.

Muschaweck, J.

Muschaweck, J. A.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” in Design of Efficient Illumination Systems, R. J. Koshel, ed. (SPIE, 2003), p. 133.

Pan, J.-W.

Petrovic, N.

N. Petrovic, I. Cohen, B. J. Frey, R. Koetter, and T. S. Huang, “Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models,” in 2001 IEEE conference on computer vision and pattern recognition (IEEE Comput. Soc, 2001), I-743–I-748.

Pohl, W.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” in Design of Efficient Illumination Systems, R. J. Koshel, ed. (SPIE, 2003), p. 133.

Raskar, R.

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of the tenth IEEE international conference on computer vision, ICCV 2005 (IEEE, 2005), 174–181 Vol. 1.

Ries, H.

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

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” in Design of Efficient Illumination Systems, R. J. Koshel, ed. (SPIE, 2003), p. 133.

Riser, A. P.

W. J. Cassarly, S. R. David, D. G. Jenkins, A. P. Riser, and T. L. R. Davenport, “Automated design of a uniform distribution using faceted reflectors,” Opt. Eng 39(7), 1830 (2000).
[Crossref]

Rolland, J. P.

Russell, R. D.

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

Sasián, J. M.

L. Wang, J. M. Sasián, P. Su, and R. J. Koshel, “Generation of uniform illumination using faceted reflectors,” in Nonimaging Optics: Efficient Design for Illumination and Solar Concentration VI, R. Winston and J. M. Gordon, eds. (SPIE, 2009), 74230Y.

Schreiber, P.

P. Schreiber, S. Kudaev, P. Dannberg, and U. D. Zeitner, “Homogeneous LED-illumination using microlens arrays,” in Optics & Photonics 2005, R. Winston and R. J. Koshel, eds. (SPIE, 2005), 59420 K.

Scott, P. J.

X. Jiang and P. J. Scott, Advanced metrology. Freeform surfaces / X. Jane Jiang and Paul J. Scott, eds., (Academic Press, 2020).

Su, P.

L. Wang, J. M. Sasián, P. Su, and R. J. Koshel, “Generation of uniform illumination using faceted reflectors,” in Nonimaging Optics: Efficient Design for Illumination and Solar Concentration VI, R. Winston and J. M. Gordon, eds. (SPIE, 2009), 74230Y.

Sulman, M. M.

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

Sun, W.-S.

Timinger, A. L.

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” in Design of Efficient Illumination Systems, R. J. Koshel, ed. (SPIE, 2003), p. 133.

Tsai, M.-D.

Walker, T.

S. R. David, T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” in International Optical Design Conference 1998, L. R. Gardner and K. P. Thompson, eds. (SPIE, 1998), p. 437.

Wang, C.-M.

Wang, L.

L. Wang, J. M. Sasián, P. Su, and R. J. Koshel, “Generation of uniform illumination using faceted reflectors,” in Nonimaging Optics: Efficient Design for Illumination and Solar Concentration VI, R. Winston and J. M. Gordon, eds. (SPIE, 2009), 74230Y.

Williams, J. F.

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

Xiang, H.

Xu, L.

Zeitner, U. D.

P. Schreiber, S. Kudaev, P. Dannberg, and U. D. Zeitner, “Homogeneous LED-illumination using microlens arrays,” in Optics & Photonics 2005, R. Winston and R. J. Koshel, eds. (SPIE, 2005), 59420 K.

Zheng, Z.-r.

Zhenrong, Z.

Appl. Numer. Math. (1)

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

Appl. Opt. (3)

Comput. Sci. Eng. (1)

J. D. Hunter, “Matplotlib: A 2D Graphics Environment,” Comput. Sci. Eng. 9(3), 90–95 (2007).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Eng (1)

W. J. Cassarly, S. R. David, D. G. Jenkins, A. P. Riser, and T. L. R. Davenport, “Automated design of a uniform distribution using faceted reflectors,” Opt. Eng 39(7), 1830 (2000).
[Crossref]

Opt. Express (3)

OSA Continuum (1)

Other (10)

J. Chaves, Introduction to nonimaging optics (CRC Press, 2008).

W. Pohl, C. Anselm, C. Knoflach, A. L. Timinger, J. A. Muschaweck, and H. Ries, “Complex 3D-tailored facets for optimal lighting of facades and public places,” in Design of Efficient Illumination Systems, R. J. Koshel, ed. (SPIE, 2003), p. 133.

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of the tenth IEEE international conference on computer vision, ICCV 2005 (IEEE, 2005), 174–181 Vol. 1.

M. Essameldin, F. Fleischmann, and T. Henning, “Problems of using the PMA adaptive mesh method in lens-array design for LED signal lighting,” in David Kidger, (Ed.), 2018 – Illuminaton Optics V, p. 25.

N. Petrovic, I. Cohen, B. J. Frey, R. Koetter, and T. S. Huang, “Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models,” in 2001 IEEE conference on computer vision and pattern recognition (IEEE Comput. Soc, 2001), I-743–I-748.

“Zemax-LLC, \Zemax Opticstudio,” http://www.zemax.com/os/opticstudio .

P. Schreiber, S. Kudaev, P. Dannberg, and U. D. Zeitner, “Homogeneous LED-illumination using microlens arrays,” in Optics & Photonics 2005, R. Winston and R. J. Koshel, eds. (SPIE, 2005), 59420 K.

L. Wang, J. M. Sasián, P. Su, and R. J. Koshel, “Generation of uniform illumination using faceted reflectors,” in Nonimaging Optics: Efficient Design for Illumination and Solar Concentration VI, R. Winston and J. M. Gordon, eds. (SPIE, 2009), 74230Y.

S. R. David, T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” in International Optical Design Conference 1998, L. R. Gardner and K. P. Thompson, eds. (SPIE, 1998), p. 437.

X. Jiang and P. J. Scott, Advanced metrology. Freeform surfaces / X. Jane Jiang and Paul J. Scott, eds., (Academic Press, 2020).

Supplementary Material (2)

NameDescription
» Visualization 1       This experiment evaluates the influence of partial illumination. Changing the active aperture area is performed by varying the Iris diameter from 1 mm to 28 mm, then detecting the illuminance pattern corresponding to each aperture.
» Visualization 2       This experiment investigates the optical performance of the freeform lens-array compared to the traditional one-to-one freeform lens.

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Figures (10)

Fig. 1.
Fig. 1. (a) Beam-shaping, considering the one-to-one design approach, and the light energy at a distant target, and (b) Dividing the light source energy into parts, then redistributing and superimposing the light intensity using freeform lens-lets creating a structured illuminance
Fig. 2.
Fig. 2. (a) The image of the Town Musicians of Bremen, (b) Uniform grid representing the light source collimated irradiance, and (c) Adaptive grid representing the required mapping
Fig. 3.
Fig. 3. Simulation setup
Fig. 4.
Fig. 4. (a) Surface gradients, (b) Sag data of one lens-let, and (c) Sag data of the lens-array
Fig. 5.
Fig. 5. (a) Simulation setup, (b) Input irradiance, and (c) Simulation results
Fig. 6.
Fig. 6. (a) Fabricated freeform lens-array, (b) Experimental setup including the collimated beam, fabricated freeform lens-array, and blank camera sensor, and (c) Experimental results
Fig. 7.
Fig. 7. Three different light sources: (a) Irradiance according to the datasheet, (b) Irradiance measured using Goniophotometer, (c) Created illuminance patterns [Camera black-red scale], (d) Error correlation [Jet-Colormap [20]]
Fig. 8.
Fig. 8. (a) Schematic diagram (Iris diaphragm), (b) Created illuminance pattern correspond to each diameter, (c) Error Correlation, and (c) Root mean square values (see Visualization 1)
Fig. 9.
Fig. 9. Superimposition of all illuminance patterns
Fig. 10.
Fig. 10. (a) Freeform lens – Sag data and Simulation results, (b) Freeform lens-array – Sag data and Simulation results, and (c) RMS values of error correlation (see Visualization 2).

Equations (10)

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I S ( ξ ) d ξ = I T ( x ) d x ,
ξ = ( ξ , η ) T , and x = ( x , y ) T .
det 2 Ψ ( ξ ) I T ( Ψ ( ξ ) ) = I S ( ξ ) ,
d Ψ d t = log ( det 2 Ψ ( ξ ) I T ( Ψ ( ξ ) ) I S ( ξ ) ) ,
I S ( ξ ) = 1.
S ( x , y ) = ( S x , S y ) T ,
n 1 ( i x n ) = n 2 ( r x n ) ,
z f = ( x 2 + y 2 200 ) ,
RMS = i = 1 n a i 2 n ,
Final illuminance pattern = i = 1 n p i ,