Abstract

Computer-generated phase-only holograms can be used for laser beam shaping, i.e., for focusing a given aperture with intensity and phase distributions into a pregiven intensity pattern in their focal planes. A numerical approach based on iterative finite-element mesh adaption permits the design of appropriate phase functions for the task of focusing into two-dimensional reconstruction patterns. Both the hologram aperture and the reconstruction pattern are covered by mesh mappings. An iterative procedure delivers meshes with intensities equally distributed over the constituting elements. This design algorithm adds new elementary focuser functions to what we call object-oriented hologram design. Some design examples are discussed.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1978), Vol. 16, pp. 119–232.
    [CrossRef]
  2. J. C. Dainty, Laser Speckle and Related Phenomena (Springer, Berlin, 1984).
  3. L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Method of making an object-dependent diffusor,” U.S. Patent3,619,022 (9November1971).
  4. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–264 (1972).
  5. R. Bräuer, F. Wyrowski, O. Bryngdahl, “Diffusors in digital holography,” J. Opt. Soc. Am. A 8, 572–578 (1991).
    [CrossRef]
  6. P. W. Rhodes, D. L. Shealy, “Refractive optical system for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980).
    [CrossRef] [PubMed]
  7. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965).
    [CrossRef]
  8. N. C. Roberts, “Beam shaping by holographic filters,” Appl. Opt. 28, 31–32 (1989).
    [CrossRef] [PubMed]
  9. C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian laser beam to a near field uniform beam,” Opt. Eng. 30, 537–543 (1991).
    [CrossRef]
  10. Z. Jaroszewicz, A. Kolodziejczyk, D. Mouriz, S. Bara, “Analytic design of computer-generated Fourier-transform holograms for plane curves reconstruction,” J. Opt. Soc. Am. A 8, 559–565 (1991).
    [CrossRef]
  11. M. A. Golub, I. N. Sisakyan, V. A. Soifer, “Infrared radiation focusators,” Opt. Lasers Eng. 15, 297–309 (1991).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  13. T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
    [CrossRef] [PubMed]
  14. J. E. Akin, Finite Elements for Analysis and Design (Academic, London, 1994).
  15. H. R. Schwarz, Finite Element Methods (Academic, London, 1980).
  16. B. Jaehne, Digital Image Processing (Springer, Berlin, 1991).
  17. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
  18. H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
    [CrossRef]
  19. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990).
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1996 (1)

1994 (1)

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

1991 (4)

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian laser beam to a near field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Z. Jaroszewicz, A. Kolodziejczyk, D. Mouriz, S. Bara, “Analytic design of computer-generated Fourier-transform holograms for plane curves reconstruction,” J. Opt. Soc. Am. A 8, 559–565 (1991).
[CrossRef]

M. A. Golub, I. N. Sisakyan, V. A. Soifer, “Infrared radiation focusators,” Opt. Lasers Eng. 15, 297–309 (1991).
[CrossRef]

R. Bräuer, F. Wyrowski, O. Bryngdahl, “Diffusors in digital holography,” J. Opt. Soc. Am. A 8, 572–578 (1991).
[CrossRef]

1989 (1)

1980 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–264 (1972).

1965 (1)

Akin, J. E.

J. E. Akin, Finite Elements for Analysis and Design (Academic, London, 1994).

Aleksoff, C. C.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian laser beam to a near field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Bara, S.

Beyerlein, M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Bräuer, R.

Bryngdahl, O.

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer, Berlin, 1984).

Dresel, T.

Ellis, K. K.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian laser beam to a near field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Falkenstörfer, O.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990).

Frieden, B. R.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–264 (1972).

Golub, M. A.

M. A. Golub, I. N. Sisakyan, V. A. Soifer, “Infrared radiation focusators,” Opt. Lasers Eng. 15, 297–309 (1991).
[CrossRef]

Goodman, J. W.

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

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Method of making an object-dependent diffusor,” U.S. Patent3,619,022 (9November1971).

Jaehne, B.

B. Jaehne, Digital Image Processing (Springer, Berlin, 1991).

Jaroszewicz, Z.

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Method of making an object-dependent diffusor,” U.S. Patent3,619,022 (9November1971).

Kolodziejczyk, A.

Lee, W.-H.

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Method of making an object-dependent diffusor,” U.S. Patent3,619,022 (9November1971).

Lindlein, N.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Mouriz, D.

Neagle, B. D.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian laser beam to a near field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990).

Rhodes, P. W.

Roberts, N. C.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–264 (1972).

Schwarz, H. R.

H. R. Schwarz, Finite Element Methods (Academic, London, 1980).

Schwider, J.

T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
[CrossRef] [PubMed]

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Shealy, D. L.

Sickinger, H.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Sisakyan, I. N.

M. A. Golub, I. N. Sisakyan, V. A. Soifer, “Infrared radiation focusators,” Opt. Lasers Eng. 15, 297–309 (1991).
[CrossRef]

Soifer, V. A.

M. A. Golub, I. N. Sisakyan, V. A. Soifer, “Infrared radiation focusators,” Opt. Lasers Eng. 15, 297–309 (1991).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Wyrowski, F.

Appl. Opt. (4)

J. Opt. Soc. Am. A (2)

Opt. Eng. (2)

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian laser beam to a near field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Opt. Lasers Eng. (1)

M. A. Golub, I. N. Sisakyan, V. A. Soifer, “Infrared radiation focusators,” Opt. Lasers Eng. 15, 297–309 (1991).
[CrossRef]

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–264 (1972).

Other (10)

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

J. C. Dainty, Laser Speckle and Related Phenomena (Springer, Berlin, 1984).

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Method of making an object-dependent diffusor,” U.S. Patent3,619,022 (9November1971).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990).

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

J. E. Akin, Finite Elements for Analysis and Design (Academic, London, 1994).

H. R. Schwarz, Finite Element Methods (Academic, London, 1980).

B. Jaehne, Digital Image Processing (Springer, Berlin, 1991).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (20)

Fig. 1
Fig. 1

Reconstruction geometry: An incident wave on the aperture plane E is to be focused into a pregiven intensity pattern at the focal plane F.

Fig. 2
Fig. 2

Node numbering for a square mesh.

Fig. 3
Fig. 3

Four-point boundary neighborhood in a square mesh.

Fig. 4
Fig. 4

Mesh mapping from a square to a circular aperture obtained by four-point interpolation.

Fig. 5
Fig. 5

Every inner mesh node P 0 is surrounded by four nearest neighbors P 1, …, P 4. The point C denotes their center of gravity.

Fig. 6
Fig. 6

Mesh mapping, optimized by Laplace iteration, from a square to a circular aperture.

Fig. 7
Fig. 7

Eight-point neighborhood of an inner mesh node P 0.

Fig. 8
Fig. 8

Mesh mapping from a square to a circular aperture. The net has been optimized so that all quadrilaterals have the same area.

Fig. 9
Fig. 9

Reconstruction pattern that resembles the topology of a square mesh. Again, all quadrilaterals have the same area.

Fig. 10
Fig. 10

Notation for a triangular mesh.

Fig. 11
Fig. 11

Six-point boundary neighborhood in a triangular mesh.

Fig. 12
Fig. 12

Mesh mapping of a triangle to a circular aperture after six-point interpolation and Laplace iteration.

Fig. 13
Fig. 13

Nine-point neighborhood used to obtain triangles with equal areas.

Fig. 14
Fig. 14

Mesh mapping of a triangle to a circular aperture, according to Eqs. (24) and (25).

Fig. 15
Fig. 15

Real part of a phase-only hologram function belonging to the nets shown in Figs. 2 and 8.

Fig. 16
Fig. 16

Simulated reconstruction modulus of the hologram shown in Fig. 15.

Fig. 17
Fig. 17

Real part of a phase-only hologram function belonging to the net shown in Fig. 9.

Fig. 18
Fig. 18

Simulated reconstruction modulus of the hologram shown in Fig. 17.

Fig. 19
Fig. 19

Real part of a phase-only hologram function belonging to the nets shown in Figs. 10 and 14.

Fig. 20
Fig. 20

Simulated reconstruction modulus of the hologram shown in Fig. 19.

Equations (35)

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

k Ψ ( u ) = Φ H ( u ) + ϕ E ( u ) .
Ψ ( u ) = x ( u ) - u f ,
E = E I E ( u ) d 2 u = F I F ( x ) d 2 x .
I E ( u ) d 2 u = I F ( x ) d 2 x ,
I F [ x ( u ) ] det x u = I E ( u ) .
x i j = - β + 2 β i I - 1 ,
y i j = - β + 2 β j J - 1 ,
P 0 : = l = 1 4 g l P l ,
g 1 = d 2 d 3 d 4 ( d 1 + d 2 ) ( d 1 d 2 + d 3 d 4 ) , g 2 = d 1 d 3 d 4 ( d 1 + d 2 ) ( d 1 d 2 + d 3 d 4 ) , g 3 = d 1 d 2 d 4 ( d 3 + d 4 ) ( d 3 d 4 + d 1 d 2 ) , g 4 = d 1 d 2 d 3 ( d 3 + d 4 ) ( d 3 d 4 + d 1 d 2 ) .
C : = 1 4 ( P 1 + P 2 + P 3 + P 4 ) .
P 0 : = P 0 + α D ,
2 x i 2 + 2 x j 2 = 0 ,
2 y i 2 + 2 y j 2 = 0 ,
A 1 + A 2 = A 3 + A 4 ,
A 1 + A 3 = A 2 + A 4 ,
x c ( y 2 - y 8 - y 6 + y 4 ) - y c ( x 2 - x 8 - x 6 + x 4 ) = x 1 ( y 2 - y 8 ) + x 5 ( y 4 - y 6 ) + y 1 ( x 8 - x 2 ) + y 5 ( x 6 - x 4 ) ,
x c ( y 8 - y 6 - y 4 + y 2 ) - y c ( x 8 - x 6 - x 4 + x 2 ) = x 3 ( y 2 - y 4 ) + x 7 ( y 8 - y 6 ) + y 3 ( x 4 - x 2 ) + y 7 ( x 6 - x 8 ) .
P 0 : = l = 1 6 g l P l ,
g 1 = d 2 d 3 d 4 d 5 d 6 ( d 1 + d 2 ) ( d 1 d 2 + d 3 d 4 ) ( d 1 d 2 + d 5 d 6 ) , g 2 = d 1 d 3 d 4 d 5 d 6 ( d 1 + d 2 ) ( d 1 d 2 + d 3 d 4 ) ( d 1 d 2 + d 5 d 6 ) , g 3 = d 1 d 2 d 4 d 5 d 6 ( d 3 + d 4 ) ( d 3 d 4 + d 1 d 2 ) ( d 3 d 4 + d 5 d 6 ) , g 4 = d 1 d 2 d 3 d 5 d 6 ( d 3 + d 4 ) ( d 3 d 4 + d 1 d 2 ) ( d 3 d 4 + d 5 d 6 ) , g 5 = d 1 d 2 d 3 d 4 d 6 ( d 5 + d 6 ) ( d 5 d 6 + d 1 d 2 ) ( d 5 d 6 + d 3 d 4 ) , g 6 = d 1 d 2 d 3 d 4 d 5 ( d 5 + d 6 ) ( d 5 d 6 + d 1 d 2 ) ( d 5 d 6 + d 3 d 4 ) .
A + C + D + H + I = ! A + C + B + E + F ,
E + F + G + H + I = ! A + C + B + E + F ,
D + H + I = ! B + E + F ,
G + H + I = ! A + C + B .
x c ( y 5 + y 6 - y 2 - y 9 ) + y c ( x 2 + x 9 - x 5 - x 6 ) x 7 ( y 6 - y 8 ) + y 7 ( x 8 - x 6 ) + x 4 ( y 5 - y 3 ) + y 4 ( x 3 - x 5 ) + x 9 y 8 - y 9 x 8 + x 2 y 3 - x 3 y 2 ,
x c ( y 6 + y 8 - y 2 - y 3 ) + y c ( x 2 + x 3 - x 6 - x 8 ) x 1 ( y 9 - y 2 ) + y 1 ( x 2 - x 9 ) + x 4 ( y 5 - y 3 ) + y 4 ( x 3 - x 5 ) + x 6 y 5 - y 5 x 6 + x 9 y 8 - x 8 y 9 .
Ψ ( u , v ) = k = 0 D l = 0 D - k a k l u k v l ,
m : = k 2 D - k + 3 2 + l + 1 , a m : = a k l , Ω m ( u , v ) : = u k v l .
Ψ ( u , v ) = m = 2 M a m Ω m ( u , v ) ,
Ψ u = m = 2 M a m Ω m u ,
Ψ v = m = 2 M a m Ω m v .
g ( a ) : = i , j { [ m = 2 M a m Ω ( u i j , v i j ) u - ψ u i j ) 2 + [ m = 2 M a m Ω ( u i j , v i j ) v - ψ v i j ] 2 } .
g ( a ) a m = ! 0 ,             for m { 2 , , M } .
m = 2 M c n m a m = b n ,             n { 2 , , M } ,
c m n = i , j [ Ω m ( u i j , v i j ) u Ω n ( u i j , v i j ) u + Ω m ( u i j , v i j ) v Ω n ( u i j , v i j ) v ] ,
b n = i , j [ ψ u i j Ω n ( u i j , v i j ) u + ψ v i j Ω n ( u i j , v i j ) v ] .

Metrics