Abstract

A new numrical method is described for analysis of the imaging properties of microlenses. This wave-propagation method is compared with the classical beam-propagation method from which it is derived. The applicability of the two methods is given and demonstrated by examples. The beam-propagation method is fast but is applicable only for small apertures; the new wave-propagation method requires no paraxial approximation but requires more computational effort.

© 1993 Optical Society of America

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References

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  1. K.-H. Brenner, “Three-dimensional integration of digital optical systems,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 25–28.
  2. K.-H. Brenner, W. Eckert, S. Kufner, S. Sinzinger, G. Borghs, M. Kuijk, P. Heremans, “Cascading of two pnpn-photothyristor arrays in a micro-optical system:” in Diffractive Optics: Design, Fabrication, and Applications, Vol.9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 3–9.
  3. M. Oikawa, K. Iga, “Distributed-index planar microlens,” Appl. Opt. 21, 1052–1056 (1982).
    [CrossRef] [PubMed]
  4. J. Jahns, K.-H. Brenner, W. Däschner, C. Doubrava, T. Merklein, “Replication of diffractive micro-optical elements using a PMMA molding technique,” Optik 89, 98–100 (1992)
  5. Z. D. Popovic, R. A. Sprague, G. A. N. Connell, “Technique for monolithic fabrication of microlens arrays,” Appl. Opt. 27, 1281–1284 (1988).
    [CrossRef] [PubMed]
  6. M. C. Hutley, “Optical techniques for the generation of microlens arrays,” J. Mod. Opt. 37, 253–265 (1990).
    [CrossRef]
  7. M. Frank, M. Kufner, S. Kufner, M. Testorf, “Microlenses in poly(methyl methacrylate) with high relative aperture,” Appl. Opt. 30, 2666–2667 (1991).
    [CrossRef] [PubMed]
  8. J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  9. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  10. S. Misawa, K. Iga, “Estimation of planar microlens by oblique ray tracing,” Appl. Opt. 27, 480–485 (1988).
    [CrossRef] [PubMed]
  11. A. Ishikawa, M. Izutsu, T. Sueta, “Beam propagation method analysis of optical waveguide lenses,” Appl. Opt. 29, 5064–5068 (1990).
    [CrossRef] [PubMed]
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 147–149.

1992 (1)

J. Jahns, K.-H. Brenner, W. Däschner, C. Doubrava, T. Merklein, “Replication of diffractive micro-optical elements using a PMMA molding technique,” Optik 89, 98–100 (1992)

1991 (1)

1990 (2)

M. C. Hutley, “Optical techniques for the generation of microlens arrays,” J. Mod. Opt. 37, 253–265 (1990).
[CrossRef]

A. Ishikawa, M. Izutsu, T. Sueta, “Beam propagation method analysis of optical waveguide lenses,” Appl. Opt. 29, 5064–5068 (1990).
[CrossRef] [PubMed]

1988 (2)

1982 (1)

1978 (1)

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Borghs, G.

K.-H. Brenner, W. Eckert, S. Kufner, S. Sinzinger, G. Borghs, M. Kuijk, P. Heremans, “Cascading of two pnpn-photothyristor arrays in a micro-optical system:” in Diffractive Optics: Design, Fabrication, and Applications, Vol.9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 3–9.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 147–149.

Brenner, K.-H.

J. Jahns, K.-H. Brenner, W. Däschner, C. Doubrava, T. Merklein, “Replication of diffractive micro-optical elements using a PMMA molding technique,” Optik 89, 98–100 (1992)

K.-H. Brenner, “Three-dimensional integration of digital optical systems,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 25–28.

K.-H. Brenner, W. Eckert, S. Kufner, S. Sinzinger, G. Borghs, M. Kuijk, P. Heremans, “Cascading of two pnpn-photothyristor arrays in a micro-optical system:” in Diffractive Optics: Design, Fabrication, and Applications, Vol.9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 3–9.

Connell, G. A. N.

Däschner, W.

J. Jahns, K.-H. Brenner, W. Däschner, C. Doubrava, T. Merklein, “Replication of diffractive micro-optical elements using a PMMA molding technique,” Optik 89, 98–100 (1992)

Doubrava, C.

J. Jahns, K.-H. Brenner, W. Däschner, C. Doubrava, T. Merklein, “Replication of diffractive micro-optical elements using a PMMA molding technique,” Optik 89, 98–100 (1992)

Eckert, W.

K.-H. Brenner, W. Eckert, S. Kufner, S. Sinzinger, G. Borghs, M. Kuijk, P. Heremans, “Cascading of two pnpn-photothyristor arrays in a micro-optical system:” in Diffractive Optics: Design, Fabrication, and Applications, Vol.9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 3–9.

Feit, M. D.

M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fleck, J. A.

M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Frank, M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.

Heremans, P.

K.-H. Brenner, W. Eckert, S. Kufner, S. Sinzinger, G. Borghs, M. Kuijk, P. Heremans, “Cascading of two pnpn-photothyristor arrays in a micro-optical system:” in Diffractive Optics: Design, Fabrication, and Applications, Vol.9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 3–9.

Hutley, M. C.

M. C. Hutley, “Optical techniques for the generation of microlens arrays,” J. Mod. Opt. 37, 253–265 (1990).
[CrossRef]

Iga, K.

Ishikawa, A.

Izutsu, M.

Jahns, J.

J. Jahns, K.-H. Brenner, W. Däschner, C. Doubrava, T. Merklein, “Replication of diffractive micro-optical elements using a PMMA molding technique,” Optik 89, 98–100 (1992)

Kufner, M.

Kufner, S.

M. Frank, M. Kufner, S. Kufner, M. Testorf, “Microlenses in poly(methyl methacrylate) with high relative aperture,” Appl. Opt. 30, 2666–2667 (1991).
[CrossRef] [PubMed]

K.-H. Brenner, W. Eckert, S. Kufner, S. Sinzinger, G. Borghs, M. Kuijk, P. Heremans, “Cascading of two pnpn-photothyristor arrays in a micro-optical system:” in Diffractive Optics: Design, Fabrication, and Applications, Vol.9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 3–9.

Kuijk, M.

K.-H. Brenner, W. Eckert, S. Kufner, S. Sinzinger, G. Borghs, M. Kuijk, P. Heremans, “Cascading of two pnpn-photothyristor arrays in a micro-optical system:” in Diffractive Optics: Design, Fabrication, and Applications, Vol.9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 3–9.

Merklein, T.

J. Jahns, K.-H. Brenner, W. Däschner, C. Doubrava, T. Merklein, “Replication of diffractive micro-optical elements using a PMMA molding technique,” Optik 89, 98–100 (1992)

Misawa, S.

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Oikawa, M.

Popovic, Z. D.

Sinzinger, S.

K.-H. Brenner, W. Eckert, S. Kufner, S. Sinzinger, G. Borghs, M. Kuijk, P. Heremans, “Cascading of two pnpn-photothyristor arrays in a micro-optical system:” in Diffractive Optics: Design, Fabrication, and Applications, Vol.9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 3–9.

Sprague, R. A.

Sueta, T.

Testorf, M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 147–149.

Appl. Opt. (6)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

J. Mod. Opt. (1)

M. C. Hutley, “Optical techniques for the generation of microlens arrays,” J. Mod. Opt. 37, 253–265 (1990).
[CrossRef]

Optik (1)

J. Jahns, K.-H. Brenner, W. Däschner, C. Doubrava, T. Merklein, “Replication of diffractive micro-optical elements using a PMMA molding technique,” Optik 89, 98–100 (1992)

Other (4)

K.-H. Brenner, “Three-dimensional integration of digital optical systems,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 25–28.

K.-H. Brenner, W. Eckert, S. Kufner, S. Sinzinger, G. Borghs, M. Kuijk, P. Heremans, “Cascading of two pnpn-photothyristor arrays in a micro-optical system:” in Diffractive Optics: Design, Fabrication, and Applications, Vol.9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 3–9.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 147–149.

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Figures (10)

Fig. 1
Fig. 1

Definition of the coordinate system and the propagation angle.

Fig. 2
Fig. 2

Ewald-sphere for different refractive indices. The transversal components of the k vector are the same, but the length and consequently the z component changes.

Fig. 3
Fig. 3

Illustration of the change of propagation direction at an interface according to Huygens’ construction.

Fig. 4
Fig. 4

Theoretical model for determination of the angle of refraction by different simulation methods.

Fig. 5
Fig. 5

Comparison of the angle of refraction obtained by Snell’s law (crosses), time BPM (squares), and the WPM (circles).

Fig. 6
Fig. 6

Illustration of Maxwell’s fisheye. Each point on the radius r0 is perfectly focused on the opposite point.

Fig. 7
Fig. 7

Simulation of light propagation through Maxwell’s fisheye with the BPM. The diameter of the lens is 340 μm. The focus is shifted to shorter distances and shows spherical aberration.

Fig. 8
Fig. 8

Simulation of light propagation through Maxwell’s fisheye with the WPM. The focal distance is correct, and the intensity distribution is almost symmetrical.

Fig. 9
Fig. 9

(a) Phase profile behind a hyperbolical step-index lens of 270-μm diameter, (b) phase difference relative to an ideal sphere.

Fig. 10
Fig. 10

Parallel simulation of the light propagation of five plane waves through a step-index lens of 300-μm diameter.

Equations (11)

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

E ( x , y , z + δ z ) = exp { - i δ z [ 2 + n ( x , y , z ) 2 k 0 2 ] 1 / 2 } × E ( x , y , z ) .
h ( ν x , z + δ z ) = ( ν x , z ) exp { i δ z [ n ¯ ( z ) 2 k 0 2 - ( 2 π ν x ) 2 ] 1 / 2 } ,
λ 0 ν x = n ¯ ( z ) sin ( ϑ ¯ ) = [ n ¯ ( z ) + δ n ( x ) ] sin ( ϑ ¯ + δ ϑ ¯ ) .
δ ϑ δ n ( x ) n ¯ ( z ) tan ( ϑ ¯ ) .
E ( x , z + δ z ) = E h ( x , z + δ z ) exp [ - i δ n ( x ) k 0 δ z ] .
δ ϕ 1 - cos ( ϑ ) .
E ( x , z ) = - ( ν x , z ) exp ( 2 π i ν x x ) d ν x .
exp ( - i k z δ z ) = exp [ - i n ( x , y ) k 0 cos ( ϑ ) δ z ] ,
n ( x , y ) sin ( ϑ ) = λ 0 ν x .
E ( x , z + δ z ) = - ( ν x , z ) exp ( 2 π i ν x x ) × exp [ - i n ( x , y ) k 0 cos ( ϑ ) δ z ] d ν x .
δ z λ δ n .

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