Abstract

Picture-generating freeform surfaces are able to generate a picture in a defined plane by incoherent beam shaping comparable to illumination purposes. No classical imaging is performed. Therefore the classical Rayleigh criterion of the diffraction limit cannot be applied. In this paper, we investigate the physical light formation of picture-generating freeform surfaces using Fresnel-Huygens-based simulations. A criterion for the diffraction limit was found. The resolution of such surfaces is significantly inferior to the resolution of classical imaging systems. However, in many cases, such systems are limited by the geometrical resolution. The influence of those two limitations were examined and a maximum of resolution, being limited by diffraction and by geometrical parameters can be found.

© 2012 OSA

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Errata

S. Zwick, R. Feßler, J. Jegorov, and G. Notni, "Erratum: Resolution limitations for tailored picture-generating freeform surfaces," Opt. Express 20, 26743-26743 (2012)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-24-26743

References

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    [CrossRef]
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    [CrossRef] [PubMed]
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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]
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2011

D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems.” Opt. Lett. 36, 918–920 (2011).
[CrossRef] [PubMed]

S. Zwick, P. Kühmstedt, and G. Notni, “Phase-shifting fringe projection system using freeform optics,” Proc. SPIE 8169, 816933 (2011).

2010

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2010).
[CrossRef]

2009

B. Yang, J. Makinen, M. Aikio, G. Jin, and Y. Wang, “Free-form lens design for wide-angle imaging with an equidistance projection scheme,” Optik 120, 74–78 (2009).
[CrossRef]

M. Kurz, D. Oberschmidt, N. Siedow, R. Fessler, and J. Jegorovs, “Mit schnellem Algorithmus zur perfekten Freiformoptik,” Mikroproduktion 03, 10–12 (2009).

2008

2002

D. L. Shealy, “Optical design of laser beam shaping systems,” Proc. SPIE 4832, 344–358 (2002).
[CrossRef]

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

1992

E. Adelson and J. Wang, “Single lens stereo with a plenoptic camera,” IEEE Transactions on Pattern Analysis and Machine Intelligence 14, 99–106 (1992).
[CrossRef]

1971

R. Shack and B. Platt, “Abstract: Production and Use of a Lenticular Hartmann Screen,” J. Opt. Soc. Am. 61, 656 (1971).

Adelson, E.

E. Adelson and J. Wang, “Single lens stereo with a plenoptic camera,” IEEE Transactions on Pattern Analysis and Machine Intelligence 14, 99–106 (1992).
[CrossRef]

Aikio, M.

B. Yang, J. Makinen, M. Aikio, G. Jin, and Y. Wang, “Free-form lens design for wide-angle imaging with an equidistance projection scheme,” Optik 120, 74–78 (2009).
[CrossRef]

Benitez, P.

Benítez, P.

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2010).
[CrossRef]

Blechinger, F.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol. 3 (Wiley-VCH, 2007).

Blen, J.

Bräuer, A.

D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems.” Opt. Lett. 36, 918–920 (2011).
[CrossRef] [PubMed]

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Brennesholtz, M.

M. Brennesholtz and E. Stupp, Projection Displays (Wiley SID, 2008).

Duval, G.

R. Ng, M. Levoy, G. Duval, M. Horowitz, and P. Hanrahan, “Light Field Photography with a Hand-held Plenoptic Camera,” Stanford Tech Report CTSR 2005-02 (2005).

Fessler, R.

M. Kurz, D. Oberschmidt, N. Siedow, R. Fessler, and J. Jegorovs, “Mit schnellem Algorithmus zur perfekten Freiformoptik,” Mikroproduktion 03, 10–12 (2009).

Gebhardt, A.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gross, H.

H. Gross, Handbook of Optical Systems, Vol. 1 (Wiley-VCH, 2005).
[CrossRef]

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol. 3 (Wiley-VCH, 2007).

Hanrahan, P.

R. Ng, M. Levoy, G. Duval, M. Horowitz, and P. Hanrahan, “Light Field Photography with a Hand-held Plenoptic Camera,” Stanford Tech Report CTSR 2005-02 (2005).

Hicks, R. A.

Horowitz, M.

R. Ng, M. Levoy, G. Duval, M. Horowitz, and P. Hanrahan, “Light Field Photography with a Hand-held Plenoptic Camera,” Stanford Tech Report CTSR 2005-02 (2005).

Jegorovs, J.

M. Kurz, D. Oberschmidt, N. Siedow, R. Fessler, and J. Jegorovs, “Mit schnellem Algorithmus zur perfekten Freiformoptik,” Mikroproduktion 03, 10–12 (2009).

Jin, G.

B. Yang, J. Makinen, M. Aikio, G. Jin, and Y. Wang, “Free-form lens design for wide-angle imaging with an equidistance projection scheme,” Optik 120, 74–78 (2009).
[CrossRef]

Kudaev, S.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Kühmstedt, P.

S. Zwick, P. Kühmstedt, and G. Notni, “Phase-shifting fringe projection system using freeform optics,” Proc. SPIE 8169, 816933 (2011).

Kurz, M.

M. Kurz, D. Oberschmidt, N. Siedow, R. Fessler, and J. Jegorovs, “Mit schnellem Algorithmus zur perfekten Freiformoptik,” Mikroproduktion 03, 10–12 (2009).

Levoy, M.

R. Ng, M. Levoy, G. Duval, M. Horowitz, and P. Hanrahan, “Light Field Photography with a Hand-held Plenoptic Camera,” Stanford Tech Report CTSR 2005-02 (2005).

Makinen, J.

B. Yang, J. Makinen, M. Aikio, G. Jin, and Y. Wang, “Free-form lens design for wide-angle imaging with an equidistance projection scheme,” Optik 120, 74–78 (2009).
[CrossRef]

Michaelis, D.

D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems.” Opt. Lett. 36, 918–920 (2011).
[CrossRef] [PubMed]

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Minano, J. C.

Miñano, J. C.

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2010).
[CrossRef]

Muschaweck, J.

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

Ng, R.

R. Ng, M. Levoy, G. Duval, M. Horowitz, and P. Hanrahan, “Light Field Photography with a Hand-held Plenoptic Camera,” Stanford Tech Report CTSR 2005-02 (2005).

Notni, G.

S. Zwick, P. Kühmstedt, and G. Notni, “Phase-shifting fringe projection system using freeform optics,” Proc. SPIE 8169, 816933 (2011).

Novikov, S. P.

A. V. Pogorelov and S. P. Novikov, Multidimensional Monge-Ampere equation (Harwood Academic Publishers, 1995).

Oberschmidt, D.

M. Kurz, D. Oberschmidt, N. Siedow, R. Fessler, and J. Jegorovs, “Mit schnellem Algorithmus zur perfekten Freiformoptik,” Mikroproduktion 03, 10–12 (2009).

Oliker, V.

V. Oliker, “Mathematical Aspects of Design of Beam Shaping Surfaces in Geometrical Optics,” in Trends in Nonlinear Analysis, M. Kirkilionis, S. Kromker, R. Rannacher, and F. Tomi eds. (Springer, 2003), pp. 193–224.

Peschka, M.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol. 3 (Wiley-VCH, 2007).

Platt, B.

R. Shack and B. Platt, “Abstract: Production and Use of a Lenticular Hartmann Screen,” J. Opt. Soc. Am. 61, 656 (1971).

Pogorelov, A. V.

A. V. Pogorelov and S. P. Novikov, Multidimensional Monge-Ampere equation (Harwood Academic Publishers, 1995).

Ries, H.

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

Santamaria, A.

Santamaría, A.

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2010).
[CrossRef]

Schreiber, P.

D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems.” Opt. Lett. 36, 918–920 (2011).
[CrossRef] [PubMed]

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Shack, R.

R. Shack and B. Platt, “Abstract: Production and Use of a Lenticular Hartmann Screen,” J. Opt. Soc. Am. 61, 656 (1971).

Shealy, D. L.

D. L. Shealy, “Optical design of laser beam shaping systems,” Proc. SPIE 4832, 344–358 (2002).
[CrossRef]

Siedow, N.

M. Kurz, D. Oberschmidt, N. Siedow, R. Fessler, and J. Jegorovs, “Mit schnellem Algorithmus zur perfekten Freiformoptik,” Mikroproduktion 03, 10–12 (2009).

Steinkopf, R.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

Stupp, E.

M. Brennesholtz and E. Stupp, Projection Displays (Wiley SID, 2008).

Voelz, D. G.

D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE Tutorial Texts Vol. TT89) (SPIE Press, 2011).

Wang, J.

E. Adelson and J. Wang, “Single lens stereo with a plenoptic camera,” IEEE Transactions on Pattern Analysis and Machine Intelligence 14, 99–106 (1992).
[CrossRef]

Wang, Y.

B. Yang, J. Makinen, M. Aikio, G. Jin, and Y. Wang, “Free-form lens design for wide-angle imaging with an equidistance projection scheme,” Optik 120, 74–78 (2009).
[CrossRef]

Winston, R.

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier Academic Press, 2005).

Yang, B.

B. Yang, J. Makinen, M. Aikio, G. Jin, and Y. Wang, “Free-form lens design for wide-angle imaging with an equidistance projection scheme,” Optik 120, 74–78 (2009).
[CrossRef]

Zügge, H.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol. 3 (Wiley-VCH, 2007).

Zwick, S.

S. Zwick, P. Kühmstedt, and G. Notni, “Phase-shifting fringe projection system using freeform optics,” Proc. SPIE 8169, 816933 (2011).

IEEE Transactions on Pattern Analysis and Machine Intelligence

E. Adelson and J. Wang, “Single lens stereo with a plenoptic camera,” IEEE Transactions on Pattern Analysis and Machine Intelligence 14, 99–106 (1992).
[CrossRef]

J. Opt. Soc. Am.

R. Shack and B. Platt, “Abstract: Production and Use of a Lenticular Hartmann Screen,” J. Opt. Soc. Am. 61, 656 (1971).

J. Opt. Soc. Am. A.

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

Mikroproduktion

M. Kurz, D. Oberschmidt, N. Siedow, R. Fessler, and J. Jegorovs, “Mit schnellem Algorithmus zur perfekten Freiformoptik,” Mikroproduktion 03, 10–12 (2009).

Opt. Express

Opt. Lett.

Opt. Rev.

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2010).
[CrossRef]

Optik

B. Yang, J. Makinen, M. Aikio, G. Jin, and Y. Wang, “Free-form lens design for wide-angle imaging with an equidistance projection scheme,” Optik 120, 74–78 (2009).
[CrossRef]

Proc. SPIE

D. L. Shealy, “Optical design of laser beam shaping systems,” Proc. SPIE 4832, 344–358 (2002).
[CrossRef]

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905 (2008).
[CrossRef]

S. Zwick, P. Kühmstedt, and G. Notni, “Phase-shifting fringe projection system using freeform optics,” Proc. SPIE 8169, 816933 (2011).

Other

A. V. Pogorelov and S. P. Novikov, Multidimensional Monge-Ampere equation (Harwood Academic Publishers, 1995).

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol. 3 (Wiley-VCH, 2007).

H. Gross, Handbook of Optical Systems, Vol. 1 (Wiley-VCH, 2005).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE Tutorial Texts Vol. TT89) (SPIE Press, 2011).

M. Brennesholtz and E. Stupp, Projection Displays (Wiley SID, 2008).

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier Academic Press, 2005).

V. Oliker, “Mathematical Aspects of Design of Beam Shaping Surfaces in Geometrical Optics,” in Trends in Nonlinear Analysis, M. Kirkilionis, S. Kromker, R. Rannacher, and F. Tomi eds. (Springer, 2003), pp. 193–224.

F. M. Dickey and S. C. Holswade, eds., Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).
[CrossRef]

R. Ng, M. Levoy, G. Duval, M. Horowitz, and P. Hanrahan, “Light Field Photography with a Hand-held Plenoptic Camera,” Stanford Tech Report CTSR 2005-02 (2005).

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Figures (8)

Fig. 1
Fig. 1

Color online: Tailored freeform lens generating a picture of the Fraunhofer emblem.

Fig. 2
Fig. 2

Color online: Projector systems.

Fig. 3
Fig. 3

Huygens-Fresnel principle applied to a freeform mirror (for reasons of clearity, the optical path has been depicted unfolded).

Fig. 4
Fig. 4

Color online: Minimal feature size Δx for freeform surfaces with varying number of displayed periods N. The Rayleigh line resolution is plotted for comparison.

Fig. 5
Fig. 5

Color online: Propagation of an intensity pattern generated by a freeform surface. The freeform surface has been designed to generate a sinusoidal pattern at a distance of 50 mm. At a distance of around z=100 mm, diffraction-limited spots are generated (for reasons of clearity, the intensity at z=100 mm has been scaled by a factor of 0.25))

Fig. 6
Fig. 6

Color online: Subapertures (SA) on the freeform surface generate the picture as a coherent superposition of the light distributions generated by each subaperture separately. (a) Cross section of a freeform surface with N=4 showing the subapertures. (b) Intensity distribution resulting from light propagation for the whole freeform (SA1 to SA4), for single subapertures separately (SA2 and SA3) and coherent superposition (SA2+SA3).

Fig. 7
Fig. 7

Color online: Image close to the diffraction limit according to Sparrow. The sinusoidal line pattern is strongly disturbed by diffraction. However, the four main maxima are still visible.

Fig. 8
Fig. 8

Color online: There are two opposing tendencies for diffraction limited and geometrical resolution. Therefore depending on the geometrical arrangement (z, Mff), an optimum of resolution can be found.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

Ψ : entrance surface of L S .
Δ x Point Rayleigh = 0.61 λ NA
Δ x Line Rayleigh = 0.5 λ NA
U ( P l ) = 1 i λ D F F U ( F m ) e i k r F P r F P cos θ d x d y
U ( F m ) = e i k r Q F r Q F .
I ( P l ) | U ( P l ) | 2 = | D F F e i k ( r Q F + r F P ) r Q F r F P d x d y | 2 .
NA sub = NA N = D f f / N 2 z = D sub 2 z
Δ x S = 0.733 λ N A sub .
Δ x geom = M f f D sub = 2 z D picture D f f 2 z N A sub

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