Abstract

One of the major problems in freeform illumination design in a geometrical optics approximation is picture generation with extended light sources. In contrast to the freeform design with zero-étendue sources, the extension of the light source leads to the typical blurring effect and a contrast reduction of the required irradiance. This effect can be minimized by increasing the distance between the freeform surface and the light source, which according to étendue conservation, results in an impractically large projection optic. To tackle this problem, we propose a design concept consisting of the combination of a pattern-generating double freeform surface for collimated beam shaping, which is calculated for a zero-étendue light source, and an imaging projection system with a telecentric object space. The design concept works independently of the shape of the emission area of the light source and does not require a representation of the extended light source by several individual wavefronts. By interpreting the pattern blurring effect as a composition of a shift contribution and a distortion contribution, we show that both can be minimized simultaneously by an appropriate placement of the object plane of the imaging optics and by making the distance between both freeform surfaces as small as possible. This allows the calculation of compact, energy-efficient freeform illumination systems for picture generation with real extended light sources. We demonstrate the significant blurring reduction by designing a simple illumination system consisting of a collimation optic, a (zero-étendue) double freeform lens for collimated beam shaping, and a projection lens for the generation of the target distribution “Elaine” with an extended Lambertian emitter of 3mm×3mm extension and ±42deg maximum opening angle. For a working distance to the projection system of 500 mm and a target area of 300mm×300mm, a relative blurring extension of 2% is estimated, compared to 23% for a single freeform projector with the same energy throughput and similar lateral extension. The influence of the doublefreeform thickness on the blurring reduction is demonstrated, and a summary of the design procedure for the developed design concept is given.

© 2019 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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  4. V. I. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express 25, A58–A72 (2017).
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  5. S. Zwick, P. Kühmstedt, and G. Notni, “Phase-shifting fringe projection system using freeform optics,” Proc. SPIE 8169, 81690W (2011).
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    [Crossref]
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    [Crossref]
  8. O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
    [Crossref]
  9. Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
    [Crossref]
  10. 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]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  14. R. Wu and H. Hua, “Direct design of aspherical lenses for extended non-Lambertian sources in three-dimensional rotational geometry,” Opt. Express 24, 1017–1030 (2016).
    [Crossref]
  15. A. Aksoylar, “Modelling and model-aware signal processing methods for enhancement of optical systems,” Ph.D. dissertation (Boston University, 2016).
  16. M. Brand and A. Aksoylar, “Sharp images from freeform optics and extended light sources,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2016), paper FW5H.2.
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  22. C. Bösel and H. Gross, “Single freeform surface design for prescribed input wavefront and target irradiance,” J. Opt. Soc. Am. A 34, 1490–1499 (2017).
    [Crossref]
  23. C. Bösel and H. Gross, “Double freeform illumination design for prescribed wavefronts and irradiances,” J. Opt. Soc. Am. A 35, 236–243 (2018).
    [Crossref]
  24. C. Bösel, J. Hartung, and H. Gross, “Irradiance and phase control with two freeform surfaces using partial differential equations,” Proc. SPIE 10693, 106930C (2018).
    [Crossref]
  25. L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26, 24602–24613 (2018).
    [Crossref]
  26. V. Oliker, L. L. Doskolovich, and D. A. Bykov, “Beam shaping with a plano-freeform lens pair,” Opt. Express 26, 19406–19419 (2018).
    [Crossref]
  27. N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, “Computation of double freeform optical surfaces using a Monge-Ampère solver: application to beam shaping,” Opt. Commun. 439, 251–259 (2019).
    [Crossref]

2019 (1)

N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, “Computation of double freeform optical surfaces using a Monge-Ampère solver: application to beam shaping,” Opt. Commun. 439, 251–259 (2019).
[Crossref]

2018 (5)

2017 (4)

2016 (1)

2015 (3)

2014 (1)

2013 (2)

2011 (2)

S. Zwick, P. Kühmstedt, and G. Notni, “Phase-shifting fringe projection system using freeform optics,” Proc. SPIE 8169, 81690W (2011).
[Crossref]

J. Bortz and N. Shatz, “Relationships between the generalized functional method and other methods of nonimaging optical design,” Appl. Opt. 50, 1488–1500 (2011).
[Crossref]

2010 (2)

2008 (1)

2004 (2)

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]

O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[Crossref]

2002 (1)

Aksoylar, A.

A. Aksoylar, “Modelling and model-aware signal processing methods for enhancement of optical systems,” Ph.D. dissertation (Boston University, 2016).

M. Brand and A. Aksoylar, “Sharp images from freeform optics and extended light sources,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2016), paper FW5H.2.

Andreev, E. S.

Benítez, P.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photon. Rev. 12, 1700310 (2018).
[Crossref]

S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Miñano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25, 29627–29641 (2017).
[Crossref]

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]

O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[Crossref]

Berens, M.

Bezus, E. A.

Blen, J.

S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Miñano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25, 29627–29641 (2017).
[Crossref]

O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[Crossref]

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]

Bortz, J.

Bösel, C.

Brand, M.

M. Brand and A. Aksoylar, “Sharp images from freeform optics and extended light sources,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2016), paper FW5H.2.

Bräuer, A.

D. Michaelis, P. Schreiber, C. Li, A. Bräuer, and H. Gross, “Freeform array projection,” Proc. SPIE 9629, 962909 (2015).
[Crossref]

Bräuer-Burchardt, C.

Brix, K.

Bykov, D. A.

Cassarly, W. J.

Chaves, J.

S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Miñano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25, 29627–29641 (2017).
[Crossref]

O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[Crossref]

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]

Ding, Y.

Doskolovich, L. L.

Dross, O.

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]

O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[Crossref]

Duerr, F.

Falicoff, W.

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]

Feng, Z.

Fournier, F. R.

Grabovickic, D.

Gross, H.

C. Bösel and H. Gross, “Double freeform illumination design for prescribed wavefronts and irradiances,” J. Opt. Soc. Am. A 35, 236–243 (2018).
[Crossref]

C. Bösel, J. Hartung, and H. Gross, “Irradiance and phase control with two freeform surfaces using partial differential equations,” Proc. SPIE 10693, 106930C (2018).
[Crossref]

C. Bösel and H. Gross, “Single freeform surface design for prescribed input wavefront and target irradiance,” J. Opt. Soc. Am. A 34, 1490–1499 (2017).
[Crossref]

D. Michaelis, P. Schreiber, C. Li, A. Bräuer, and H. Gross, “Freeform array projection,” Proc. SPIE 9629, 962909 (2015).
[Crossref]

Gu, P. F.

Hafizogullari, Y.

Han, Y.

Hartung, J.

C. Bösel, J. Hartung, and H. Gross, “Irradiance and phase control with two freeform surfaces using partial differential equations,” Proc. SPIE 10693, 106930C (2018).
[Crossref]

Heist, S.

Herkommer, A. M.

D. Rausch, M. Rommel, A. M. Herkommer, and T. Talpur, “Illumination design for extended sources based on phase space mapping,” Opt. Eng. 56, 065103 (2017).
[Crossref]

Hernández, M.

S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Miñano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25, 29627–29641 (2017).
[Crossref]

O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[Crossref]

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]

Hua, H.

Huber, S.

IJzerman, W. L.

N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, “Computation of double freeform optical surfaces using a Monge-Ampère solver: application to beam shaping,” Opt. Commun. 439, 251–259 (2019).
[Crossref]

Krause, S.

Kühmstedt, P.

Li, C.

D. Michaelis, P. Schreiber, C. Li, A. Bräuer, and H. Gross, “Freeform array projection,” Proc. SPIE 9629, 962909 (2015).
[Crossref]

Li, H.

Liang, R.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photon. Rev. 12, 1700310 (2018).
[Crossref]

D. Ma, Z. Feng, and R. Liang, “Deconvolution method in designing freeform lens array for structured light illumination,” Appl. Opt. 54, 1114–1117 (2015).
[Crossref]

Liu, P.

Liu, X.

Loosen, P.

Luo, Y.

Ma, D.

Michaelis, D.

D. Michaelis, P. Schreiber, C. Li, A. Bräuer, and H. Gross, “Freeform array projection,” Proc. SPIE 9629, 962909 (2015).
[Crossref]

Miñano, J. C.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photon. Rev. 12, 1700310 (2018).
[Crossref]

S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Miñano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25, 29627–29641 (2017).
[Crossref]

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]

O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[Crossref]

Mohedano, R.

S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Miñano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25, 29627–29641 (2017).
[Crossref]

O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[Crossref]

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]

Müller, G.

Muñoz, F.

O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[Crossref]

Muschaweck, J.

Notni, G.

Oliker, V.

Oliker, V. I.

Platen, A.

Rausch, D.

D. Rausch, M. Rommel, A. M. Herkommer, and T. Talpur, “Illumination design for extended sources based on phase space mapping,” Opt. Eng. 56, 065103 (2017).
[Crossref]

Ries, H.

Rolland, J. P.

Rommel, M.

D. Rausch, M. Rommel, A. M. Herkommer, and T. Talpur, “Illumination design for extended sources based on phase space mapping,” Opt. Eng. 56, 065103 (2017).
[Crossref]

Schreiber, P.

D. Michaelis, P. Schreiber, C. Li, A. Bräuer, and H. Gross, “Freeform array projection,” Proc. SPIE 9629, 962909 (2015).
[Crossref]

Shatz, N.

Sorgato, S.

Steinkopf, R.

Stollenwerk, J.

Talpur, T.

D. Rausch, M. Rommel, A. M. Herkommer, and T. Talpur, “Illumination design for extended sources based on phase space mapping,” Opt. Eng. 56, 065103 (2017).
[Crossref]

ten Thije Boonkkamp, J. H. M.

N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, “Computation of double freeform optical surfaces using a Monge-Ampère solver: application to beam shaping,” Opt. Commun. 439, 251–259 (2019).
[Crossref]

Thienpont, H.

Völl, A.

Wester, R.

Wu, R.

Xu, L.

Yadav, N. K.

N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, “Computation of double freeform optical surfaces using a Monge-Ampère solver: application to beam shaping,” Opt. Commun. 439, 251–259 (2019).
[Crossref]

Zhang, Y.

Zheng, Z.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photon. Rev. 12, 1700310 (2018).
[Crossref]

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

Zheng, Z. R.

Zwick, S.

Appl. Opt. (3)

J. Opt. Soc. Am. A (4)

Laser Photon. Rev. (1)

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of freeform illumination optics,” Laser Photon. Rev. 12, 1700310 (2018).
[Crossref]

Opt. Commun. (1)

N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, “Computation of double freeform optical surfaces using a Monge-Ampère solver: application to beam shaping,” Opt. Commun. 439, 251–259 (2019).
[Crossref]

Opt. Eng. (2)

D. Rausch, M. Rommel, A. M. Herkommer, and T. Talpur, “Illumination design for extended sources based on phase space mapping,” Opt. Eng. 56, 065103 (2017).
[Crossref]

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]

Opt. Express (9)

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
[Crossref]

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]

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 18, 9055–9063 (2010).
[Crossref]

R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22, A552–A560 (2014).
[Crossref]

V. Oliker, L. L. Doskolovich, and D. A. Bykov, “Beam shaping with a plano-freeform lens pair,” Opt. Express 26, 19406–19419 (2018).
[Crossref]

L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26, 24602–24613 (2018).
[Crossref]

S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Miñano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25, 29627–29641 (2017).
[Crossref]

R. Wu and H. Hua, “Direct design of aspherical lenses for extended non-Lambertian sources in three-dimensional rotational geometry,” Opt. Express 24, 1017–1030 (2016).
[Crossref]

V. I. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express 25, A58–A72 (2017).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (4)

O. Dross, R. Mohedano, P. Benítez, J. C. Miñano, J. Chaves, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[Crossref]

D. Michaelis, P. Schreiber, C. Li, A. Bräuer, and H. Gross, “Freeform array projection,” Proc. SPIE 9629, 962909 (2015).
[Crossref]

C. Bösel, J. Hartung, and H. Gross, “Irradiance and phase control with two freeform surfaces using partial differential equations,” Proc. SPIE 10693, 106930C (2018).
[Crossref]

S. Zwick, P. Kühmstedt, and G. Notni, “Phase-shifting fringe projection system using freeform optics,” Proc. SPIE 8169, 81690W (2011).
[Crossref]

Other (2)

A. Aksoylar, “Modelling and model-aware signal processing methods for enhancement of optical systems,” Ph.D. dissertation (Boston University, 2016).

M. Brand and A. Aksoylar, “Sharp images from freeform optics and extended light sources,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2016), paper FW5H.2.

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Figures (12)

Fig. 1.
Fig. 1. Ray path in an optical system between the surfaces z 0 ( x ) and z N + 1 ( x ) . The ray path is defined by the input direction (vector field) s 0 ( x ) and the surface intersection coordinates x i ( x ) , i = 1 , , N + 1 .
Fig. 2.
Fig. 2. Illustration of the scaling properties of the pattern blurring for single freeform illumination design. The single freeform z F F ( x ) is calculated for a point light source and an illumination plane z = z T . Every point of the freeform surface is thereby hit by exactly one ray of the zero-étendue light source. Then, a pinhole is placed at one point of the freeform surface, and the corresponding source ray (black) is considered. If an extended light source is assumed, the pinhole images a certain maximum range of source ray directions, denoted by the red rays. Hence, the extension of the light source causes a blurring extension Δ T around the ideal target point of the zero-étendue light source. The blurred pattern corresponds to a superposition of all pinhole contributions of every freeform surface point. Therefore, to minimize the blurring extension Δ T for a fixed target size, the freeform surface should be calculated for a target plane z = z T , which is as close as possible to z F F ( x ) .
Fig. 3.
Fig. 3. Freeform illumination system generates the required pattern on a specific intermediate plane z = z T interm . Due to the light source extension, the distance between the intermediate plane and the freeform should be as small as possible for minimal blurring. The intermediate illumination pattern on z = z T interm is then considered as the object of the subsequent imaging system with a telecentric object space, whereby the object cone angle of the imaging system is defined by the maximum angle between the rays on z = z T interm and the z axis. The intermediate illumination pattern is then imaged to the target plane z = z T with the required magnification. Since the imaging system only magnifies the intermediate pattern, the relative blurring on the intermediate and target plane will be identical.
Fig. 4.
Fig. 4. Freeform surface is calculated for (a) a point source located at (0,0). (b) A shifted point source located at ( Δ P L S , x , 0 ) generates a pattern, which is shifted by Δ x interm and distorted. (c) An extended light source generates a blurred pattern, which is the superposition of every point source contribution.
Fig. 5.
Fig. 5. Comparison of a (a) single freeform design for irradiance control and a (b) double freeform design for irradiance and wavefront control. In both designs, there is a shift Δ x interm due to the residual angle α of the collimated input beam. In (b), this shift is compensated for by an appropriate placement of the object plane of the imaging system.
Fig. 6.
Fig. 6. (a) Prescribed irradiance “Elaine”. (b) Projection system geometry. (c) Ray tracing result for an ideal collimated beam with α = 0 .
Fig. 7.
Fig. 7. (a) Single freeform surface for the pattern generation and its (d) ray tracing result. (b) Single freeform surface with projection lens for the pattern generation and its (e) ray tracing result. (c) Double freeform surface with projection lens for the pattern generation and its (f) ray tracing result. The pattern blurring is significantly reduced by introduction of the projection lens and double freeform surface.
Fig. 8.
Fig. 8. Influence of the double freeform thickness on the pattern blurring. The system geometry corresponds to Fig. 6(b). All three freeform lenses have the same lateral extension, which, according to étendue conservation, makes a comparison of the degree of blurring reasonable. Irradiance pattern for a thickness of (a)  0.2 arb. units, (b)  0.4 arb. units, and (c)  0.8 arb. units. If the double freeform thickness is too large, there is no advantage regarding the blurring compared to a single freeform combined with a projection lens.
Fig. 9.
Fig. 9. (a) System geometry. The distance between the projection system and the target plane is 500 mm. The target area is 300 mm × 300 mm . (b) Freeform projection system consisting of a rotational symmetric collimation lens, a double freeform, and a rotational symmetric projection lens. The light source is a 3 mm × 3 mm Lambertian emitter with a maximum opening angle of ± 42 . (c) Irradiance distribution generated by the projection system.
Fig. 10.
Fig. 10. (a) Single freeform surface with and extended source. The source extension causes a shift Δ x interm in the intermediate plane z = z T interm = z OP for every point source contribution. With a (b) double freeform surface for irradiance and phase control, this shift can be compensated by an appropriate placement of the object plane z = z OP of the projection lens.
Fig. 11.
Fig. 11. (a) Thick double freeform lens and projection lens. The double freeform surface is calculated for a PLS placed at (0,0) and a collimated output beam. The PLS is shifted to ( Δ P L S , x , 0 ) . (b) Irradiance at z = z T from ray tracing simulation with the object plane z = z OP placed between both freeform surfaces. The irradiance “Elaine” can be observed two times due to the influence of the irradiance and wavefront shaping by both lenses. Depending on if the object plane is closer to the first or second freeform, one of the patterns will become more or less visible.
Fig. 12.
Fig. 12. Freeform array channel geometry according to Ref. [20]. The double freeform is calculated for a collimated input beam and a spherical converging output wavefront with the focal point at z = z P L . The reference plane for the irradiance is denoted by z = z T interm . The size of the irradiance therefore scales with the distance to z = z P L . To compensate for the offset Δ x interm by Δ OP , x , the object plane z = z OP has to be placed appropriately. Due to the scaling of the irradiance Δ x interm and Δ OP , x are not equal.

Equations (11)

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x N + 1 ( x 0 ) = i = 0 N z i + 1 ( x i + 1 ) z i ( x i ) ( s ^ i + 1 ) z ( x i ) ( ( s ^ i + 1 ) x ( x i ) ( s ^ i + 1 ) y ( x i ) ) + x 0 ,
s ^ i + 1 ( x i ) = n i n i 1 s ^ i ( x i 1 ) + { n i n i 1 · n ^ i ( x i ) · s ^ i ( x i 1 ) + 1 ( n i n i 1 ) 2 [ 1 ( n ^ i ( x i ) · s ^ i ( x i 1 ) ) 2 ] } n ^ i ( x i ) , s ^ i + 1 ( x i ) = s ^ i ( x i 1 ) 2 [ n ^ i ( x i ) · s ^ i ( x i 1 ) ] n ^ i ( x i ) ,
n i ( x i ) = ( x i z i ( x i ) 1 ) , x ( x , y ) T
x i z i ( x i ) = 1 n i 1 ( s ^ i ) z ( x i 1 ) n i ( s ^ i + 1 ) z ( x i ) · ( n i 1 ( s ^ i ) x ( x i 1 ) n i ( s ^ i + 1 ) x ( x i ) n i 1 ( s ^ i ) y ( x i 1 ) n i ( s ^ i + 1 ) y ( x i ) ) , i = 1 , , N ,
s i + 1 ( x i ) = ( x i + 1 x i y i + 1 y i z i + 1 ( x i + 1 ) z i ( x i ) ) , i = 1 , , N .
det ( u ( x ) ) I T ( u ( x ) ) = I S ( x ) ,
x j z F F , S ( x ) = f ( x z F F , S ( x ) , y z F F , S ( x ) )
u ( x ) = f ( z F F , S ( x ) , x z F F , S ( x ) , y z F F , S ( x ) , x j + 1 ( x ) , , x N ( x ) ) , det ( u ( x ) ) I T ( u ( x ) ) = I S ( x ) , B C : u ( Ω S ) = Ω T
n 1 n 2 · z F F , I I ( x ) z F F , I ( x )
Δ T 2 Δ x interm α · z T z F F ( x ) ,
Δ T 2 Δ x interm α · n 1 / n 2 · z F F , I I ( x ) z F F , I ( x ) .

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