Abstract

Image formation of three-dimensional objects suffers from out-of-focus noise if the light is coherent. Nevertheless, it is possible to generate two noiseless image intensities (IA and IB) at two depth locations by means of a single computer hologram. The phases related to IA and IB provide design freedom. To accomplish this goal, we use a ping-pong algorithm that bounces back and fourth between the two planes. Our ping-pong algorithm is the fourth member of a family of algorithms. The first member is known as the Gerchberg–Saxton algorithm. Basic considerations and experimental results are presented. Details of the algorithm are explained in Appendix A.

© 1994 Optical Society of America

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References

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  1. A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik 51, 105–117 (1978).
  2. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  3. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik 35, 237–246 (1972).
  4. J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297–305 (1980).
  5. P. van Toorn, H. A. Ferwerda, “The problem of phase retrieval in light and electron microscopy of strong objects,” Opt. Acta 23, 469–481 (1976).
    [Crossref]
  6. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [Crossref] [PubMed]
  7. W. J. Dallas, “Computer generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer, New York, 1980), Chap. 6.

1990 (1)

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297–305 (1980).

1978 (1)

A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik 51, 105–117 (1978).

1976 (1)

P. van Toorn, H. A. Ferwerda, “The problem of phase retrieval in light and electron microscopy of strong objects,” Opt. Acta 23, 469–481 (1976).
[Crossref]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik 35, 237–246 (1972).

Dallas, W. J.

W. J. Dallas, “Computer generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer, New York, 1980), Chap. 6.

Ferwerda, H. A.

P. van Toorn, H. A. Ferwerda, “The problem of phase retrieval in light and electron microscopy of strong objects,” Opt. Acta 23, 469–481 (1976).
[Crossref]

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297–305 (1980).

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik 35, 237–246 (1972).

Lohmann, A. W.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik 35, 237–246 (1972).

Thomas, J. A.

van Toorn, P.

P. van Toorn, H. A. Ferwerda, “The problem of phase retrieval in light and electron microscopy of strong objects,” Opt. Acta 23, 469–481 (1976).
[Crossref]

Appl. Opt. (1)

Opt. Acta (1)

P. van Toorn, H. A. Ferwerda, “The problem of phase retrieval in light and electron microscopy of strong objects,” Opt. Acta 23, 469–481 (1976).
[Crossref]

Opt. Eng. (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297–305 (1980).

Optik (2)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik 35, 237–246 (1972).

A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik 51, 105–117 (1978).

Other (2)

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

W. J. Dallas, “Computer generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer, New York, 1980), Chap. 6.

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Figures (10)

Fig. 1
Fig. 1

Basic setup for a hologram CGH that produces intensity distributions (IA and IB) in two planes (zA and zB) separated by z0.

Fig. 2
Fig. 2

Example of two intensity distributions IA and IB as input for the CGH design.

Fig. 3
Fig. 3

Two-plane display with an image-plane CGH. The filter (FILT) eliminates unwanted diffraction orders.

Fig. 4
Fig. 4

Input intensities of (a) Heisenberg (IA) and (b) Schrödinger (IB).

Fig. 5
Fig. 5

Digital reconstructions of (a) IA and (b) IB after 50 iterations.

Fig. 6
Fig. 6

Experimental outputs (section) of the CGH after 50 iterations at (a) plane z = zA and (b) plane z = zB taken on photographic film.

Fig. 7
Fig. 7

Overall output in the Fourier plane.

Fig. 8
Fig. 8

Image IA from Fig. 6(a) and image IB from Fig. 6(b) taken by a CCD camera.

Fig. 9
Fig. 9

Fourier CGH that contains two images at two different depth locations.

Fig. 10
Fig. 10

Flow chart of the ping-pong algorithm.

Tables (1)

Tables Icon

Table 1 Four Methods with Two Specified Intensities IA(x, y) and IB(x, y) in Planes zA and zB

Equations (28)

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I A ( x , y ) = u ( x , y , z A ) 2 , u ( x , y , z A ) = A ( x , y ) exp [ i α ( x , y ) ] = u A ,
I B ( x , y ) = u ( x , y , z B ) 2 , u ( x , y , z B ) = B ( x , y ) exp [ i β ( x , y ) ] = u B .
u A ( x , y ) = δ ( x ) exp ( - i π y 2 / λ z 0 ) ,
u B ( x , y ) = δ ( y ) exp ( + i π x 2 / λ z 0 ) .
u A ( x , y ) = R A ( x ) R P ( y ) .
z T = 2 d 2 / λ .
z 0 = d 2 / 2 λ ,         3 d 2 / 2 λ , .
G A [ A K ( x ) exp ( i α K ) ] = A ( x ) exp ( i α K ) ,
G B [ B K ( x ) exp ( i β K ) ] = B ( x ) exp ( i β K ) .
A ( x ) exp [ i α 0 ( x ) ] .
F [ a exp ( i α 0 ) ] = B 1 ( x ) exp [ i β 1 ( x ) ] .
G B [ B 1 exp ( i β 1 ) ] = B ( x ) exp [ i β 1 ( x ) ] .
F - 1 [ B exp ( i β 1 ) ] = A 1 ( x ) exp [ i α 1 ( x ) ] .
G A [ A 1 exp ( i α 1 ) ] = A ( x ) exp [ i α 1 ( x ) ] .
A ( x ) exp [ i α K + 1 ( x ) ] = G A F - 1 G B F { A ( x ) exp [ i α K ( x ) ] } .
[ u K ( x ) - u K - 1 ( x ) ] 2 d x
[ A K 2 ( x ) - A K - 1 2 ( x ) ] 2 d x .
v ( x ) = u ( x ¯ ) exp [ i π ( x - x ¯ ) 2 ] d x ¯ ,
= exp ( i π x 2 ) [ u ( x ¯ ) exp ( i π x ¯ 2 ) ] exp ( - 2 π i x x ¯ ) d x ¯ .
u B ( n δ x ) = ( m ) u A ( m δ x ) exp [ + i π ( m - n ) 2 / M ] ,
= exp ( i π n 2 / M ) Σ [ u A ( m δ x ) exp ( i π m 2 / M ) ] × exp ( - 2 π i n m / M ) .
δ x = λ f / W .
Δ x / δ x = W / d .
δ z = λ ( f / W ) 2 = ( δ x ) 2 / λ .
z B - z A = z 0 = M δ z .
u B ( n δ x ) = ( m ) u A ( m δ x ) exp [ i π ( m δ x - n δ x ) 2 / λ z 0 ] .
m - n M .
propagation phase = - π λ z 0 ν 2             ( ν is a Fourier variable ) .

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