Abstract

One of the general requirements of a computer-aided design system is the existence of efficient (in data size and running time) algorithms that are generally reliable for the broadest range of design instances. The restricted data formats of the electron-beam machines impose difficulties in developing algorithms for the design of diffractive optical elements (DOE’s) and computer-generated holograms (CGH’s). Issues that are related to the development of CGH algorithms for e-beam fabrication of DOE’s and CGH’s are discussed. We define the problems the CGH algorithms need to solve, then introduce general curve drawing algorithms for the e-beam data generation of diffractive optical components. An efficient algorithm for general a spherical DOE’s is proposed. Actual design and fabrication examples are also presented.

© 1995 Optical Society of America

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References

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  1. E. R. Phillips, An Introduction to Analysis and Integration Theory (Intext Educational, Scranton, Pa., 1971), Sect. 10.4.
  2. N. M. Patrikalakis, “Surface-to-surface intersections,” IEEE Trans. Comput. Graphics Appl. 13, 89–95 (1993).
    [CrossRef]
  3. K. S. Urquhart, S. H. Lee, C. C. Guest, M. R. Feldman, H. Farhoosh, “Computer aided design of computer generated holograms for electron beam fabrication,” Appl. Opt. 28, 3387–3396 (1989).
    [CrossRef] [PubMed]
  4. S. M. Arnold, “E-beam written computer generated holograms,” Final Tech. Rep. E47069 (Honeywell Corporation Technology Center, Bloomington, Minn., 1983).
  5. D. Zaleta, B. C. Kress, W. Daschner, J. Fan, K. S. Urquhart, S. H. Lee, “Diffractive optics fabricated by electron-beam direct write methods,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 117–137.
  6. W. H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
    [CrossRef] [PubMed]
  7. S. M. Arnold, “Electron beam fabrication of computer generated holograms,” Opt. Eng. 24, 803–807 (1985).
  8. G. J. Swanson, “Binary optics technology: the theory and design of multi-level diffractive optical elements,” Tech. Rep. 854 (Lincoln Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1989).
  9. J. Logue, “Fabrication of binary optics using electron beam lithography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.884, 95–99 (1988).
  10. J. Logue, M. L. Chisholm, “General approaches to mask design for binary optics,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 19–24 (1989).
  11. H. Farhoosh, “A knowledge-based system for electron-beam fabrication of computer generated holograms,” Ph.D. dissertation (Dept. of Electrical Engineering, University of California, San Diego, Calif., 1991).
  12. T. W. Sederberg, R. J. Meyers, “Loop detection in surface patch intersections,” Comput. Aided Geom. Des. 5, 161–171 (1988).
    [CrossRef]
  13. R. E. Barnhill, S. N. Kersey, “A marching method for parametric surface/surface intersection,” Comput. Aided Geom. Des. 7, 257–280 (1990).
    [CrossRef]
  14. R. E. Barnhill, G. Farin, M. Jordan, B. R. Piper, “Surface/surface intersection,” Comput. Aided Geom. Des. 4, 285–307 (1987).
    [CrossRef]
  15. C. L. Bajai, C. M. Hoffmann, R. E. Lynch, “Tracing surface intersections,” Comput. Aided Geom. Des. 5, 3–16 (1988).
  16. D. Manocha, J. F. Canny, “A new approach for surface intersection,” Int. J. Computat. Geom. Appl. 1, 491–516.
  17. R. W. Hawley, N. C. Gallagher, “An efficient electron beam pattern data format for the production of binary computer generated holograms,” Appl. Opt. 29, 216–224 (1990).
    [CrossRef] [PubMed]
  18. D. M. Newman, R. W. Hawley, D. L. Goeckel, R. D. Crawford, S. Abraham, N. C. Gallagher, “Efficient storage, computation, and exposure of computer-generated holograms by electron-beam lithography,” Appl. Opt. 32, 2555–2565 (1993).
    [CrossRef] [PubMed]
  19. CODE V is a registered trademark of Optical Research Associates, 550 N. Rosemead Blvd., Pasadena, Calif. 91107.
  20. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical recipes in C, (Cambridge U. Press, Cambridge, 1988), Chap. 9.
  21. R. E. Bank, “Refinement algorithms and data structures for regular local mesh refinement,” in Scientific Computing, R. Stepleman et al., eds. (IMACS/North-Holland, Amsterdam, 1983), pp. 3–17.

1993

1990

R. W. Hawley, N. C. Gallagher, “An efficient electron beam pattern data format for the production of binary computer generated holograms,” Appl. Opt. 29, 216–224 (1990).
[CrossRef] [PubMed]

R. E. Barnhill, S. N. Kersey, “A marching method for parametric surface/surface intersection,” Comput. Aided Geom. Des. 7, 257–280 (1990).
[CrossRef]

1989

1988

C. L. Bajai, C. M. Hoffmann, R. E. Lynch, “Tracing surface intersections,” Comput. Aided Geom. Des. 5, 3–16 (1988).

T. W. Sederberg, R. J. Meyers, “Loop detection in surface patch intersections,” Comput. Aided Geom. Des. 5, 161–171 (1988).
[CrossRef]

1987

R. E. Barnhill, G. Farin, M. Jordan, B. R. Piper, “Surface/surface intersection,” Comput. Aided Geom. Des. 4, 285–307 (1987).
[CrossRef]

1985

S. M. Arnold, “Electron beam fabrication of computer generated holograms,” Opt. Eng. 24, 803–807 (1985).

1974

Abraham, S.

Arnold, S. M.

S. M. Arnold, “Electron beam fabrication of computer generated holograms,” Opt. Eng. 24, 803–807 (1985).

S. M. Arnold, “E-beam written computer generated holograms,” Final Tech. Rep. E47069 (Honeywell Corporation Technology Center, Bloomington, Minn., 1983).

Bajai, C. L.

C. L. Bajai, C. M. Hoffmann, R. E. Lynch, “Tracing surface intersections,” Comput. Aided Geom. Des. 5, 3–16 (1988).

Bank, R. E.

R. E. Bank, “Refinement algorithms and data structures for regular local mesh refinement,” in Scientific Computing, R. Stepleman et al., eds. (IMACS/North-Holland, Amsterdam, 1983), pp. 3–17.

Barnhill, R. E.

R. E. Barnhill, S. N. Kersey, “A marching method for parametric surface/surface intersection,” Comput. Aided Geom. Des. 7, 257–280 (1990).
[CrossRef]

R. E. Barnhill, G. Farin, M. Jordan, B. R. Piper, “Surface/surface intersection,” Comput. Aided Geom. Des. 4, 285–307 (1987).
[CrossRef]

Canny, J. F.

D. Manocha, J. F. Canny, “A new approach for surface intersection,” Int. J. Computat. Geom. Appl. 1, 491–516.

Chisholm, M. L.

J. Logue, M. L. Chisholm, “General approaches to mask design for binary optics,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 19–24 (1989).

Crawford, R. D.

Daschner, W.

D. Zaleta, B. C. Kress, W. Daschner, J. Fan, K. S. Urquhart, S. H. Lee, “Diffractive optics fabricated by electron-beam direct write methods,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 117–137.

Fan, J.

D. Zaleta, B. C. Kress, W. Daschner, J. Fan, K. S. Urquhart, S. H. Lee, “Diffractive optics fabricated by electron-beam direct write methods,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 117–137.

Farhoosh, H.

K. S. Urquhart, S. H. Lee, C. C. Guest, M. R. Feldman, H. Farhoosh, “Computer aided design of computer generated holograms for electron beam fabrication,” Appl. Opt. 28, 3387–3396 (1989).
[CrossRef] [PubMed]

H. Farhoosh, “A knowledge-based system for electron-beam fabrication of computer generated holograms,” Ph.D. dissertation (Dept. of Electrical Engineering, University of California, San Diego, Calif., 1991).

Farin, G.

R. E. Barnhill, G. Farin, M. Jordan, B. R. Piper, “Surface/surface intersection,” Comput. Aided Geom. Des. 4, 285–307 (1987).
[CrossRef]

Feldman, M. R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical recipes in C, (Cambridge U. Press, Cambridge, 1988), Chap. 9.

Gallagher, N. C.

Goeckel, D. L.

Guest, C. C.

Hawley, R. W.

Hoffmann, C. M.

C. L. Bajai, C. M. Hoffmann, R. E. Lynch, “Tracing surface intersections,” Comput. Aided Geom. Des. 5, 3–16 (1988).

Jordan, M.

R. E. Barnhill, G. Farin, M. Jordan, B. R. Piper, “Surface/surface intersection,” Comput. Aided Geom. Des. 4, 285–307 (1987).
[CrossRef]

Kersey, S. N.

R. E. Barnhill, S. N. Kersey, “A marching method for parametric surface/surface intersection,” Comput. Aided Geom. Des. 7, 257–280 (1990).
[CrossRef]

Kress, B. C.

D. Zaleta, B. C. Kress, W. Daschner, J. Fan, K. S. Urquhart, S. H. Lee, “Diffractive optics fabricated by electron-beam direct write methods,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 117–137.

Lee, S. H.

K. S. Urquhart, S. H. Lee, C. C. Guest, M. R. Feldman, H. Farhoosh, “Computer aided design of computer generated holograms for electron beam fabrication,” Appl. Opt. 28, 3387–3396 (1989).
[CrossRef] [PubMed]

D. Zaleta, B. C. Kress, W. Daschner, J. Fan, K. S. Urquhart, S. H. Lee, “Diffractive optics fabricated by electron-beam direct write methods,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 117–137.

Lee, W. H.

Logue, J.

J. Logue, M. L. Chisholm, “General approaches to mask design for binary optics,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 19–24 (1989).

J. Logue, “Fabrication of binary optics using electron beam lithography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.884, 95–99 (1988).

Lynch, R. E.

C. L. Bajai, C. M. Hoffmann, R. E. Lynch, “Tracing surface intersections,” Comput. Aided Geom. Des. 5, 3–16 (1988).

Manocha, D.

D. Manocha, J. F. Canny, “A new approach for surface intersection,” Int. J. Computat. Geom. Appl. 1, 491–516.

Meyers, R. J.

T. W. Sederberg, R. J. Meyers, “Loop detection in surface patch intersections,” Comput. Aided Geom. Des. 5, 161–171 (1988).
[CrossRef]

Newman, D. M.

Patrikalakis, N. M.

N. M. Patrikalakis, “Surface-to-surface intersections,” IEEE Trans. Comput. Graphics Appl. 13, 89–95 (1993).
[CrossRef]

Phillips, E. R.

E. R. Phillips, An Introduction to Analysis and Integration Theory (Intext Educational, Scranton, Pa., 1971), Sect. 10.4.

Piper, B. R.

R. E. Barnhill, G. Farin, M. Jordan, B. R. Piper, “Surface/surface intersection,” Comput. Aided Geom. Des. 4, 285–307 (1987).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical recipes in C, (Cambridge U. Press, Cambridge, 1988), Chap. 9.

Sederberg, T. W.

T. W. Sederberg, R. J. Meyers, “Loop detection in surface patch intersections,” Comput. Aided Geom. Des. 5, 161–171 (1988).
[CrossRef]

Swanson, G. J.

G. J. Swanson, “Binary optics technology: the theory and design of multi-level diffractive optical elements,” Tech. Rep. 854 (Lincoln Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1989).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical recipes in C, (Cambridge U. Press, Cambridge, 1988), Chap. 9.

Urquhart, K. S.

K. S. Urquhart, S. H. Lee, C. C. Guest, M. R. Feldman, H. Farhoosh, “Computer aided design of computer generated holograms for electron beam fabrication,” Appl. Opt. 28, 3387–3396 (1989).
[CrossRef] [PubMed]

D. Zaleta, B. C. Kress, W. Daschner, J. Fan, K. S. Urquhart, S. H. Lee, “Diffractive optics fabricated by electron-beam direct write methods,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 117–137.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical recipes in C, (Cambridge U. Press, Cambridge, 1988), Chap. 9.

Zaleta, D.

D. Zaleta, B. C. Kress, W. Daschner, J. Fan, K. S. Urquhart, S. H. Lee, “Diffractive optics fabricated by electron-beam direct write methods,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 117–137.

Appl. Opt.

Comput. Aided Geom. Des.

T. W. Sederberg, R. J. Meyers, “Loop detection in surface patch intersections,” Comput. Aided Geom. Des. 5, 161–171 (1988).
[CrossRef]

R. E. Barnhill, S. N. Kersey, “A marching method for parametric surface/surface intersection,” Comput. Aided Geom. Des. 7, 257–280 (1990).
[CrossRef]

R. E. Barnhill, G. Farin, M. Jordan, B. R. Piper, “Surface/surface intersection,” Comput. Aided Geom. Des. 4, 285–307 (1987).
[CrossRef]

C. L. Bajai, C. M. Hoffmann, R. E. Lynch, “Tracing surface intersections,” Comput. Aided Geom. Des. 5, 3–16 (1988).

IEEE Trans. Comput. Graphics Appl.

N. M. Patrikalakis, “Surface-to-surface intersections,” IEEE Trans. Comput. Graphics Appl. 13, 89–95 (1993).
[CrossRef]

Int. J. Computat. Geom. Appl.

D. Manocha, J. F. Canny, “A new approach for surface intersection,” Int. J. Computat. Geom. Appl. 1, 491–516.

Opt. Eng.

S. M. Arnold, “Electron beam fabrication of computer generated holograms,” Opt. Eng. 24, 803–807 (1985).

Other

G. J. Swanson, “Binary optics technology: the theory and design of multi-level diffractive optical elements,” Tech. Rep. 854 (Lincoln Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1989).

J. Logue, “Fabrication of binary optics using electron beam lithography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.884, 95–99 (1988).

J. Logue, M. L. Chisholm, “General approaches to mask design for binary optics,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 19–24 (1989).

H. Farhoosh, “A knowledge-based system for electron-beam fabrication of computer generated holograms,” Ph.D. dissertation (Dept. of Electrical Engineering, University of California, San Diego, Calif., 1991).

E. R. Phillips, An Introduction to Analysis and Integration Theory (Intext Educational, Scranton, Pa., 1971), Sect. 10.4.

S. M. Arnold, “E-beam written computer generated holograms,” Final Tech. Rep. E47069 (Honeywell Corporation Technology Center, Bloomington, Minn., 1983).

D. Zaleta, B. C. Kress, W. Daschner, J. Fan, K. S. Urquhart, S. H. Lee, “Diffractive optics fabricated by electron-beam direct write methods,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 117–137.

CODE V is a registered trademark of Optical Research Associates, 550 N. Rosemead Blvd., Pasadena, Calif. 91107.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical recipes in C, (Cambridge U. Press, Cambridge, 1988), Chap. 9.

R. E. Bank, “Refinement algorithms and data structures for regular local mesh refinement,” in Scientific Computing, R. Stepleman et al., eds. (IMACS/North-Holland, Amsterdam, 1983), pp. 3–17.

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Figures (11)

Fig. 1
Fig. 1

Fringe tracing by Trace-Ring function used in both rotationally symmetric and general aspheric fracturing algorithms. Note that ρ(t, n 1) = 0° (the e-beam angle of l 1).

Fig. 2
Fig. 2

E-beam fracturing hierarchy for rotationally symmetric DOE’s.

Fig. 3
Fig. 3

Binary masks of a spherical element generated by the Draw-Curve function: (a) EBMF, (b) MEBES.

Fig. 4
Fig. 4

Relationship between the phase error and the trace tolerance in aspheric DOE’s.

Fig. 5
Fig. 5

Algorithm for general aspheric element e-beam fabrication.

Fig. 6
Fig. 6

Picture of a typical aspherical lens fabricated by use of the aspheric algorithm. There are eight phase levels. Direct-write method is used here. Phase-error requirement is λ/50.

Fig. 7
Fig. 7

Comparison of MEBES and EBMF data sizes for rotationally symmetric DOE’s: (a) aperture size versus data sizes for both formats; (b) relationship between EBMF and MEBES is close to a constant 2.6.

Fig. 8
Fig. 8

Aspheric DOE e-beam data generated (a) by approach similar to the pixel-based methods (N = 1), (b) by N = 1 with optimization, (c) with more regions (N = 20).

Fig. 9
Fig. 9

Relationship between the number of regions and the data size for five different aspheric DOE’s. The phase-error requirement is 0.02λ.

Fig. 10
Fig. 10

Relationship between the phase error and the data size for five different aspheric DOE’s.

Fig. 11
Fig. 11

Trapezoid is split along the subfield boundaries into (a) two, (b) three, (c) four smaller pieces over an x-subfield boundary, (d) a trapezoid is split into two smaller pieces over a y-subfield boundary.

Tables (1)

Tables Icon

Table 1 Coefficients for the Five Aspheric DOE’s Designed by CODE V

Equations (7)

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δ 2 ( r j + 1 r j ) / e .
ϕ ( x , y ) = k = 0 M j = 0 k C k j x k j y j ,
ϕ ( x , y ) = ϕ 0 ,
A i = { ( x , y ) ( i 1 ) ( 2 π / m ) < ϕ ( x , y ) i ( 2 π / m ) } i .
( i 1 ) ( 2 π / m ) < ϕ ( x a , y a ) < i ( 2 π / m ) +
| Δ ϕ ( x , y ) | | ϕ ( x , y ) x Δ x + ϕ ( x , y ) y Δ y | | ϕ ( x , y ) x Δ x | + | ϕ ( x , y ) y Δ y | .
δ ( x , y ) = 2 min [ 1 / ϕ ( x , y ) x , 1 / ϕ ( x , y ) y ] ,

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